A TEXT BOOK OF HEAT FOR JUNIOR STUDENTS (INCLUDING KINETIC THEORY OF GASES, THERMODYNAMICS AND RADIATION) BY M. N. SAHA, ftSc, F.R.S., Late Palit Professor of Physics, Calcutta Universtty AND B. N, SRIVASTAVA, D.Sc^ F.N.I., Professor of General Physics, Indian Association for the Cultivation of Science, Calcutta (TWELFTH EDITION) SCIENCE BOOK AGENCY I*. I'i'-ili, Lake Terrace, CaIcutta-29 First Edition Second Impression Sf.co.nd Edition 1939 Third Edition 1943 Fourth Edition 1945 Fifth Edition 1949 Sixth Edition 1951 Seventh Edition 1953 Eighth Edition 1954 ON 1'iuntr Edition ELEVENTH !-'n Twelfth ill Rights resen Publishku by Sm, Roma Saita, ] : i-hern Avenue, Calcutta : Printed by Modern India Press 7, Raja Sudodh Mullick Square, Cai-cutta 13 Price Rupees Nine only PREFACE TO THE FIRST EDITION i Text. Book of Heat for Junior Students has been written with iv to supplying the needs of the students of the pass course ir the Bachelor's degree. It has grown out of the lectures which i he senior author has been giving to die B.Sc, pass class of Ulahabad University for several years. The plan of closely follows that of the larger Text Book which is intended for B.Sc. honours and M.Sc. students. Separate chapters have been i i voted to Kinetic Theory, Liquefaction of Gases and Heat. Engines, principles of Thermodynamics and their applications have been Created at considerable length. Throughout the book the methods of calculus have been freely employed. The supplementary chapter on orology has been kindly written by Dr. A. K. Das of the Indian Meteorological Sendee and Mr. B. N\ Srivasuava. Meteorology is a growing science and is extremely useful to die public at large. It is not at present included in the curriculum of any Indian University il Agra where it forms a special course for the M«Sc. degree) , hut this seems to be a cardinal omission. It is hoped that in future it will form a regular subject of study by degree students, \& this is the first edition of the book, it is feared that there may he several omissions and inaccuracies. The authors will be grateful if these arc brought to their notice. Allahabad : M. N, S. maty, 1933. B* N. S. PREFACE TO THE ELEVENTH EDITION Since the last edition many Indian Universities have Introduced the new three-year Degree Course while some others are still continu- ing the old two-year course. The book has therefore been thoroughly revised to cover the syllabus of the new three-year course of most- Indian universities. As this required only addition of some matter previously found in the Intermediate Syllabus and as the old subject matter of the book has been almost wholly retained, it is confidently hoped that the book will prove equally useful both for the new three- course and the old two-veai course. Calcutta : Juk t 1962. B. N. S. PREFACE TO THE TWELFTH EDITION Several suggestions for the improvement of the book, kindly i hi by teachers using the book, have been incorporated in this edition. The .standard questions, arranged chapterwise and given at the end o! the book have been brought uptndate. I. una : luajy, J 967. B, N, & ACKNOWLEDGMENTS CONTENTS We have much pleasure in expressing our indebtedness to the following authors, publishers and societies for allowing us to reproduce diagrams which appeared in the works mentioned below : — Muller-Pouillets Lehrbuch der Physik, Heading, Temperat itrmessung, Nernst, Grundlagen des neuen Warmesatzes. Jellinek, Lehrbuch der physiftalischen Chemie. Ewing, The Steam Engine and Other Heat Engines. Watson, Practical Physics. Burgess and Le Chatclicr, The Measurement of High Temperatures. Proceedings of the Physical Society of London, Physical Review. Journal of the Optical Society of America, Vol. 10. Proceedings of the Royal Society, London, Chap, II, Figs, 4, 5 ; Chap. VII, Fisr. 10. Philosophical lions of the Royal Society, Chap. II, Fi Char, vn, Fig, io. Zeiti , V, Fig, : ; , Chap, i I Fig. 16. Atmai >' ; Chap, VII, Fig. 3. [buck d< Chap, IV. Fig. 7; Chap. VI, Fig. 10; Chap. XL Fig. Handbuch der Experi physik. Chap. V, Fig. 11; Chap. VI, Pigi tap. Vni, Fig, ' hap. XT, Yi^. 11, 18, 19, 20. Ezer I (hods of Measuring Temperature. Chap. I, Figs. 5, 6, 7, 9; Chap. XT, Figs. M. 15, 21, 22, 23 from diagrams on 52, 52, 34, 71. 84, 84, 90, 90, 115 ol the work respectively. Andrade, Engines, Chap. VI, Fig. 5 ; Chap. TX, Figs. 1, 3, 4, 7, 8, IS, 16, 17 respectively from pages 239, Gl, 68, 75, 102, 86, 92, 107, 195, 211, 213 of the work by the kind permission of Messrs. G. Bell & Sons, Ltd, (London) . Partington and Shilling, Specific //eats of Gases. Chap. II, Figs. 13, 15, 17. respective:];, from pages 127, 76., B4 of the work. OH, Theory of Heat. Chap. I, Fig. 3; Chap. II, Figs. 3, 12; Chap. XI, Fig. 16, Reproduced by the kind permission ol Messrs, Macmillan & Co. Chapter Pace 1. Thermometry Temperature. Mercury thermometer. Special types of liquid thermometers. Gas thermometers, Callendar compensated air- thermometer. Standard gas thermometers. Perfect gas scale, .Jardization of secondary thermometers. Fixed temperature Platinum thei era. Measurement of resistance. mo-couples. Low temperature thermometry. International temperature scale. Illustration of the i i t Thermometry, II. Calorimetry Quantity of heat Methods in calorimetry. Method of mixtures. Radiation correction. Specific heat of solids. Specific heat o£ liquids Method of cooling. Method of melting ice. BunserVs ice calorimeter. Joly's steam calorimeter. The differential steam calorimeter. Methods baser! on the rise of temperature. oil of steady-flow electric calorimeter. Specific heat of :ter, Results of early experi- ■ arit. t. ion oi with temperature Two specific . Experiments of Gay-Lussac and Joule. Adiabatic transformations. I- ital methods. Method of mixtures, method. Measurement of C. Explosion meti Wiakitie i^xpansirn method). Experiments of Clement and Desormes. Experiments of Partington. "Ruchardt's experiment. Veloei und method. Kundt's tube. Expe- riments of Partington and Shilling. Specific heat of a incur, Results, Special calorime 1 III. Kinetic Theory of Matter The nature of heat Joule's experiments, I:. ■• "i in mics, Metl ermrning /, !••!".. ai i nnciits. i: ;>:: laboratory method of finding- /, Electrical methods. Heat as n I m decides. Growth of the kinetic tl Evidence of molecular agitation. Brownian movement Pn ted by a perfect gas. J : f gas laws. Introd of temperature. Distribution of ■. I .'.veil's law. cities. Law of equlpartition of kinetic energy. ■iilrir and atomic Mean free path phenomena. :: of the mean tree path. Transport phenomena.* Viscosity. Conduction, Value ol coi [V. Equations of State for Gases iation from the perfect gas equation. Andrews' experiments, ai der WaaU* ei of state. Methods uf finding the values of V and 'b'. Discussion of van der Waals' equation. Experimental study of the equation of state. Experimental determination of critical constants. Matter near the critical point. V. Change of State Fusion. I -a tent heat. Sublimation Change of properties on ting. Determination of the melting point. Determination of the latent heat of fusion. Experimental relationships. 104 VI Chai l*Afi» lances at liquid technique. Uses of The air conditioning' :. melting point. Fusion of cooling, boiling and superheating. Saturated ami unsati ■ lira Vaj <ur pressors of water. Statical jethods. Dynamical method. Discussion of results, Vapour pressure over curved surfaces. Latent heat of vaporization. Condensation methods. Evaporation methods. Variation of latent heat with temperature, Trouton's rule. Determination of vapour density. Accurate determination of the density of saturated vapour, VI. Production of Low Temperatures .. ,-125 Principles used in refrigeration. Adding a saft to ice. Boiling a liquid under reduced pressure. Vapour compression inachine. Refrigerants. Electric refrigerator. Ammonia absorption machine. Adiabatic expansion of compressed gas ooling due to Peltier effect. Cooling due to desorption. Liquefaction of gases. Liquefaction by application of pressure and low tempe- rature. The principle of cascade-,. Production of low tempera- tare by utilizing the Joule-Thomson effect Elementary heory of the porous iperimeui. The porous plug experiment. Principle of regenerative cooling. Lilidc's machine for liquefying air. Hampson's air h'queficr. Claude's air liqueficr, Liquefac- tion of helium. Solidification of helium. Cooling produced by adfabatic demagnetisation. Properties of helium temperatures. Low temperature liquid air. Principles of air eondi hine. VII, Thermal Expansion .. 157 ansion of solids. Lii rrlfcr measurements of liner Camparatoi method, ihod. Fizeau's interference method, fringe ter. Surface and volume expansion, Expansion of [ass, invar. Expansion of ■ bodies. Expansion of liquids. The difatometer method. The weighty thermometer mctlmd. Matthiesscn's method, Absolute expansion of Hqiuds. Hydrostatic balance method, Expansion of mercury, water. Practical app pensation of clocks and watches. Thermostats. Expansion of Rases. t Experimental determination of the volume coefficient of expansion. Experimental determination of the pressure co- efficient of expansion. VIII, Conduction of Heat . . . . . . 178- Methods of heat propagation. Conductivity of different kinds of maiter._ Definition of conductivity. Conductivity of mewls. Conductivity from calorjmetric measurement. Rectilinear flow "f heat. Mathematical investigation. Ehgen-Hausz^s experi- ment. Experiments of Despretej Wiedemann and Franz. Forbes' rcieUi I'om's metliod. Conductivity of the Earth's crust. Conduction through composite walls. Relation between the ther- mal and electrical conductivities of metals. Heat conduction in • dimensions. Conductivity of poorly conducting solids. Spherical shell method. Cylindrical shell method. Lees' disc method. Conductivity of liquids. Column method. Film method. Conductivity of gases. Hotwire method, Film method. Results. Freezing of a pond Convection of heat. Natural convection- VI 1 IPTER IK. Heat Engines Three different classes of jngines. Early history of the steam engine, bfewcomen's atmospheric engine. James Watt. fJse oi a separate condenser, The double-acting engine. Utilisation of the expansive power of steam. The governor and the throttle valve. The crank and the flywheel. Modern steam engines. Efficiency oi engines and indicator diagrams. The Carnot engine. Reversible and irreversible processes. Reversibility of the Carnot cycle Carnot's theorem. Rankhie cycle. Total heat of _ steam. Internal _ combustion i engines. The Otto cycle, 1 scl cycle. Semi-Diesel engines, The 'National' gas engine. Diesel four-stroke engines. The steam turbine, The theory of steam jets. The De Laval turbine. Ratcau and Zolly turbines, Reaction turbines (Parsons}, Alternative types of engines. Thermodynamics of refrigeratinn. Efficiency of a vapour compression machine. XI. Page 204 X. Thermodynamics . . .. 2c57 Scope oi thermodynamics. The thermal state of a body. Mathematical notes Some physical applications. Different forms of energy. Transmutation of energy. Conservation of energy. Dissipation of energy. The first law of tiicmtodynamics. Applications of the first law. _ Specific heat of a body.' Work done hi certain processes. Discontinuous changes in energy — latent heat Mess's law of constant heat-summation. Second >f thermodynamics and entropy. Scope of the second law. Preliminary statement of the second law. Absolute scale of temperature. Definition of entropy. Entropy of a syg Entropy remains constant in reversible processes. Entropy in- creases in irreversible processes. Entropy of a perfect ' gas. General statement of second law of thermodynamics. Supposed violation of the second law. Entropy and unavailable energy. Physical concept of entropy. Entropy— temperature diagrams' Entropy of steam. Applications of the two laws of thermo- dynamics, The thermodynamics! relationships (Maxwell). First on. Application to a liquid film. Second relation. Other itions. variation of intrinsic energy with volume, foute- Thontson effect. Correction of gas thermometer. Examples. Clapeyron's deduction of the Clausius-Clapevron relation, Speci- fic heat of saturated vapour. The triple point. Radiation Some simple instruments for measuring radiation. Properties and nature of radiant energy. Identity of radiant energy and hght— continuity of spectrum. Fundamental radiation processes. Theory of exchanges (Prevost), Laws of cooling. Emissive power of different substances. Reflecting power. Absorption, itions existing between the different radiation quantities Fundamental definitions, Kirchhoff's law. Applications of Kifchhoff s law, Application to astrophysics. Temperature radiation. Exchange of energy between radiation and matter in a hollow enclosure. Deduction of Kirchhoff's law. The black body. Radiometers! Sensitiveness of the thermopile. Crnokes' radiometer. Bolometer. Kadiomicrometer. Pressure of radia- tion. Total radiation from a black body— the Stcfan-Bollzmann law. Experimental verification of Stefan's law. Laws of dis- tribution of energy in blackbody spectrum. Experimental study ot the blackbody spectrum. Pyrometry. Gas pyrometers. Resis"- 278 VIII Chapter Page XII. Thermodynamics o> the ATSfosPHERE Examples (I to XII) Answers to Numerical Examples Table of Physical Constants Subject Index 316 327 331 349 351 352 CHAPTER I THERMOMETRY 1. Temperature.— The sensation of heat or cold is a matter t ail} 1 experience. By the mere sense of touch we can say whether a substance is hotter or colder than ourselves. The hot body is said to possess a higher temperature than the cold one. But the sense of touch is merely qualitative, while scientific ecision requires that every physical quantity should be measurable numerical terms. Further, the measurements must be accurate easily reproducible. This requires that the problem should be handled objectively and tile sense of touch should be discarded in favour of something which satisfies the above criteria. Let us sec how can be done. When two bodies are brought in contact, it is found that, in general., there is a change in their properties such as volume, pressure etc. due to exchange of heat. Finally an equilibrium state is attained ter which there is no further change. The two bodies are then said to be in thermal equilibrium with each other. In this state of M, I equilibrium the two bodies are said to have the same tempe- which ensures their being in thermal equilibrium, Also it is [ound that if a body A is in thermal equilibrium with two bodies B and C, then li and C will be in thermal equilibrium with each other therefore be at the same temperature. These are the two fundamental laws of thermal equilibrium and it is on account of these s that we arc able to measure the temperature of bodies B and C bringing them successively in thermal equilibrium with the thermo- neter A. The temperature of a system is a property which determirii whether or not a system is in thermal equilibrium with other systems. Heat causes many changes in the physical properties of matter some of which are well known, e.g., expansion, change in electrical .laiire. production of electromotive force at the junction of two iniilai' metals. All these effecLs have been utilised for the isurement of temperature. The earliest and commonest therm o- titilise the property of expansion. Mercury-in-glass is univets- em ployed as a thermometer for ordinary purposes, but though it iiiplc. convenient to use and direct-reading, it is not sufficiently for high-class scientific work. Mercury Thermometer.** — Everybody is familiar with the • • i ■. 1 1 1 - . 1 1 ' centigrade thermometer. It consists of a glass bulb con- mercury to which a graduated capillary stem is attached. The freezing point of water is marked 0°C and the boiling point 100°C the interval divided into 100 equal parts. This scale was first i In >1 by Celsiusf and is called the Celsius or centigrade scale I r details of construction see Preston, Theory of FIcat, Chapter 2. \ A i O -us (1701-1744) was borti at Upsala where he studied matbe- tlci ii I astronomy. In 1750 he became Professor of Astronomy and ten years hr built the observatory at Upsala and became its director. He invented '"ale. THERMOMETRY and is now adopted for all scientific work. Other stales in ordinary Use today are thos€ introduced by Fahrenheit and Reaumur. But, Fahrenheit* was the first to choose mercury as die thermoinetric substance cm account of its many advantages. It does not v, can be easily obtained pure, remains liquid over a fairly wide range, has a low specific heat and high conductivity ; it is opaque am expansion is approximately uniform and regular. But we must not orget its several drawbacks. The specific gravity and surface tension of mercury are large, and the angle of contact with glass when mercury is rising is different from that when it is falling-. On account of these defects alcohol is sometimes used in place of mercury, and since it has a larger expansion it is more sensitive but. is likely to distil over to colder pans oi the tube. The range of an ordinary mercury thermometer is limited by the fact that mercu: tes at -38.8°C and boils at 856 °C but the upper limit can be raised to about 500 "C by filling the top of the tube with nitrogen under pressure. The thermometric glass must be of special quality ; it should be stable and should rapic urn to its normal state after exposure Co high temperatures. The gla generally employed are v .■• and Jena It'. 1 " for ' her- mometer; orosilicate glass 59 111 for high-temperature work. Mercury thermomel generally o.d for rough work. If tin at all fork various corrections must be applied to get tJi: mpcrature. Tlie important ones am ihe : — This is very in po i ini sine high tempers mal ion due to change in the fundamental i om to 100-j-S (say) I i lue. ition Con his is due to want of uniformity in the bore of the capillary n (4) Correction for lag of the thermometer. This increases with !ze uf the bulb, (5) Correction due to changes in the size of the bulb caused by variable internal and external pressure, (6) Correction for the effects of capillarity. Exposed stem correction. Part of the stem and hence the contained mercury does not acquire the temperature of the bath. For details concerning the application of these corrections Appendix I. 3. Special Types of Liquid Thermometers. — The ordinary mer- cury thermometer is not suitable for certain purposes ; for this reason special types of thermometers have been devised, are "Daniel Gabriel Fahrenheit l b 16-1736) was born in Danzig of a rid fa tmstenktn. He made improvements in the t- ised his '.. ric scale. SPECIAL TYPES OF LIQUID THERMOMETERS of the ordinary thermometer designed to serve the \ lew, cological purposes thermometers arc required to ! cate the maximum and minimum temperatures to which they osed during a certain period. Six devised a combi mum and minimum thermometer which is indicated in Fig. L II if and part of die tube is filled with alcohol up to the level in with mercui") up to C above which again there is alcohol ■ glass indexes I, I, have each an iron wire attached (shown separately), and placed above B Hid CI in each tube. When the tempera- es i lie alcohol in A expands and landing mercury, on account of its tension j pushes "upwards the index ie C to its maximum limit. With a I j II in temperature this index is undis- turbed due to the viscosity of alcohol being II while the index above B is pulled 1 by the contracting alcohol, but is I. n 'hind when the temperature rises, iron wire attached to die glass in: I .: prevents it from falling under its own In. and enables its position to be I From outside. n ordinary rem -y thermometers oE I the maximum type. The stem the bulb has a constriction ■I which the mercury passes when iis temperature rises. On cooling, the iny is unable to force its way bark, I lie range of temperatures is usually. 95 °F i I I:) r and the bulb is very thin and capillary bore very fine. The mercury thread is rendered easily visible by con- . 1 1 1 1 « I i 1 1 g 1 ens-f ron t thermomc t ers. For accurate work, such as the determination of the boiling and [ting points of organic substances, several short-range thermo- • era employed between the range and 490°C. They are U thermometers. Benzol and toluol thermometer 1 1 1 1 > 1 1 • ■ ■ i the many that are in use. The Hechmann thermometer, indi I in fig. 2, is used to ni> nail changes of temperature with a high degree of accuracy. is here marked from to 5 representing approximai centigrade degrees and even? degree is di into 100 equal parts. .: reservoir at the top of the instrument, shown separately H e the range to be varied. To set the thermometer lo ii' desired range ihe bulb is heated to c n rcury into the reser- D A • • ,■}'] :r. Lib :,.. *o - ' IB 10 ' • ' V 1 10 'I 1 | , • OH - 16 i'B ■'•• ' ' i"" !-J 1 ■■'-.■ ifi. (\ E6 "j 1 r go "^^^ Fig. L — Six's maximum and imum thermometer. THERMOMETRY [CHAIV voir and the instrument gently tapped when the mercury column breaks near the reservoir and some mercury i transferred into it. Next the Beckmarm tin meter is immersed along with an auxiliary thermometer in a bath whose tempera tun varied till the mercury stands at division "l the former. The temperature corresponding to the aero of the Beckmarm thermometer is thus t -I v-i -i v e 1. 1 on 1 1 1 e a uxiliary th ermom eter, and b) varying the amount of mercury in the bulb this is adjusted to be near the desired range. The value of each scale division varies with the quantity of mercury in the bulb and a correc- tion curve Lor different settings of the zero is supplied with the instrument from which the correction at any point of the scale ma) bi obtained. 4. Gas Thermometers. — The fundamental vantage of liquid them; sters is that two thermometers containing differenl Liquids as mercury and alcohol, and graduated as on page 1 will agree probably only at.0°C and 100°C ana at no other temperature. This is due to the the expansion of the two liquid; is not regular and similar, Thus the mercury thermo- page I would give an arbitrary* 6 of temperature. Moreover, the corrections to iplied to it ■:}■■ 2) are uncertain and known ly. Hence, tor accurate work iry thermometers arc calibrated (see Sec. 8) lual comparison with a resistance tin meter throughout the entire range. Even then the Riercur) thermometer is rarely used for accurate work, and [or all standard work pas mometcrs are employed. Gases •• i/« I servants i mometric substance. Their expansion is large so that gas thermometers will be more sensitive and the expansion of the containing I will necessitate only a very small correc- tion, rheir expansion is- also regular, i.e., the nsion of a volume of gas at 0°C is the same W 2< — Becfcmauu thermometer. •The v mercury at any temperature i lying between and IQO'-'C as measured on the perfect gas scale is given by the r Pi - Bd(1 + 1-B182 x I0-'.' + 078 X 10-" t*)< - the relation ts not linear and the readings of the mercury thermometer iratod on p. I will not agree, even after applying the corrections mentioned 1 1 i A a perfect giv .. cter even in the range 0°— 100°G 1. 1 GAS rHURMOMETERS 5 cry one degree rise of temperature. They can be obtained pure and remain gaseous over wide ranges of temperature. Further, the .scales furnished by different gases are nearly identical since the volume and pressure coefficients of all permanent gases are nearly equal. Hence, gas thermometers arc used as primary standards with which -ill others are compared and calibrated. The theoretical bases underlying the use of gas as thermometric liiiii: :: are the laws of Boyle* and Charles which are very approxi- mately obeyed by the so-called permanent gases in nature but will be rigorously obeyed b] a perfect gas. Let a gas be initially at pressure /j,, volume V\ and temperature r t °C. TF we first change its pressure from p! to p^, keeping the temperature constant, and next change the temperature from t x to t 2 , keeping the pressure constant, we have n 'in these laws, . , ti' 1 + ox, where v' is the intermediate volume and a the coefficient of expansion at constant pressure, which is found experimentally to be approxi- mately equal to 1/273. Combining these two equations we have Pi v i 1 " ' i /V's l-ffcf« (1) which is the gas equation. II >'. [/a, ^2 = 0, U» at the temperature l/a = 273°C i li ice point, the volume of the gas would be zero provided the perfect gas equation is obeyed throughout the range, This temperature is, by definition, called the ah &ro. It is true ih: gas would liquefy and solidify long before this stage is reached i in I the perfect gas equation would cease to be valid. Further, it is inconceivable that matter should at any rime occupy no space (w=0). Nevertheless l of such a zero of temperature is very useful. f II we lure temperatures from this zero, the ice-point is given by I/a, the steam point by (l/«)-|-100, arid generally any tempera tun-', by (I /«) -H 2 =7YK, Th c scale so obtained is called the Kelvin • rale and will be denoted by °K, Hence equation (1) i m omes Pi"j n or ^2 J l ' where the suffix denotes the quantities at 0°C. • ■ (2) The quantity ■H Boyle (1627-1691) was b m in Trehnd but settled in England m He distinguished himsdi in the study of Physio and Chemistry and was "| i the foundation members of the Royal Society. TTis main contribution is I In' law of sases which still bears bis name. 1 I he absolute zero thus defined is shown from thermodynamic considerations to be the lowest temperature possible. Hence the idea of this absolute rry important [see further §71. : iMOMETTRY [chap. . , T is known as the gas constant and varies as the mass of the gas taken, but is approximately the sump for equivalent gram-mole- cules cl all gases. For one gram-molecule this quantity is usually denoted by R and is equal to 8.3 X ^ J " ergs/degree approximately. If the mass of the gas is increased n times, the volume at the same temperature and pressure will he increased n times and hence the gas constant will also he increased n times. Hence, the gas equation cm he. written generally in the form pv=nRT where n denotes the num- ber of gram-molecules of the gas. Equation (2) furnishes two ways of measuring temperature. The pressure may be kept constant and' the volume observed at different temperatures giving us the constant pressure thermometer ; or the volume may be kept constant and the change in pressure noted, a principle utilised in the constant volume thermometer. . Exerctee. — Calculate the pressure of 20 grams of hydrogen inside a vessel of 1 cubic metre capacity at. the temperature of 27- C. [pv=?iRT where n=2Qf c Z. Ans. 0.25 atm.] 5. Callendar Compensated Air-Thermometer. — Accurate measure- ments with the constant-pressure gas thermometers are difficult as the gas in the connecting tube and the manometer is at a tempera different from that of the bulb. To avoid this Callendar devised the compensated air thermometer. In tins instrument (Fig, 3) the pressure of the aii in the thermometer bulb B i the pressui e of the air in D as indicated by the sulphuric acid gauge G. When 11 is heated, the pressure of the air in B increases and equality of pressure is restored by allowiifg mercury to flow out from the mercury reservoir S. The volume of the tube connecting B and S is eliminated by attaching to D an exactly similar tube placed close to it. This will 'be evident from the following consideration: — Let v, v it tfg be the volume of the bulb B, the tube conned B and S P and the air in S respectively ; 6, lt a , their respective temperatures, n the number of gram-molecules of air contained in B, S and the connecting, tube anil p its pressure, we have from the gas laws, . *(t+i a-* Similarly if ?'', v/ be the volumes of the air in D and the tube attach- ed, 0', 0/ their respective temperatures, n f the number of gram- molecules and p' the pressure, we have '{r+i)-«- W STANDARD fiAS 1 MEIERS -«', * =*/, h -= &,', wc have from {3} and (4) ,i dition S and D are immersed in melting ice, a =0' =firf, reezing , water, and ine v- tf, V 2 by B, D, S B = d °-n-s (5) Mi us we see that the influence of the connecting tubes is entirely i a bed if (I) the pressure in B is kept equal to that in D (p — ff)\ i he total mass of the gas in B, S, and the connecting tube is to that in D and the connecting tube ,{n = nf) ; (3) the volumes Or tl ; ecting tubes anal (v-i — vf). The i a nd i t i o il [di—8i) is automatically satisfied i e ilie two connect- tubes are placed side by side and are at i In- same temperature. 6. Standard Gas Thermometers. — T h e i on slant-pressure a i r- thermometer has been nded by Callendar ii . arious grounds : (1) ih apparatus and the . ulations are simple ; i he internal pressure on (he bulb does not in- , as the tempera- ture vises; (3) accuracy of the results depends i ii the accuracy of hing. . tevertheless, instrument dees not give i : ■ i'. • rdant results 1 has been replaced the constan t-volume thermometer ns a stan- l. The normal ther- icter selected by the Bute au 1 ntem a ti on al pig, 4 (a ) .— ■ Constant-volume Hydrogen Thermometer, Poids et Mesures and everywhere adopted today is the constant volume hydrogen tl\er- THERMOMETRY [CHAP. meter filled witli gas at a pressure of I metre of mercury at the temperature of melting ice. It consists essentially of two parts ; the bulb enclosing the invariable gaseous mass and the manometer for measuring the pressure. Fig. 4(a) represents the thermometer dia- gr annua tieally. The bulb C is a platinum-iridium tube a litre in capacity, 1 metre in length and 3G mm. in diameter. It is attached to the manometer by a capillary tube of platinum 1 metre in length. The manometer consists of two tubes A and B, and the stem of the barometer R dips into A. The barometer tube is bent so that the upper surface of mercury in it. is exactly above \i and these levels can be read oft by a catheto- meter furnished with telescopes. The number of observations to be taken is thus reduced to two. B consists of two columns of mercury separated by the sted-picce H and both these columns communicate with A. By raising or lowering the mercury reservoir M, the mercury surface in the lower' part of B is arranged just to touch a fine platinum point P [shown separ- ately in Fig. 4(6)], projecting from the steel- piece H, and thus the volume of the enclosed gas is kept, constant. The thermometer described above is suit- :: for measuring temperatures up to 600°G, iperatures certain modifications ary which will he discussed under netry' in Chapter XI. The range of gas thermometers with Ten irons ran be extended from -200° to 1600 "C. We shall now deduce a Eormula for converting the observed pressn nga into corresponding temperatures. T.I: /?, ; „ p vn denote the pressure indicated by the manometer at the ice point and steam- point respectively, tlien p„v = riRTn, pimV — nR(T -{- 100) where T represents the ice-point on the per Feet gas scale and the fundamental interval is 100°C. Hence to *v rr ioo /to p > where 8 is the coefficient of expansion at constant volume. Thus we know ft for that gas from a measurement of p w and /_v To find absolute temperature corresponding to any observed pressure p t , we have, Tx ~Po m PERFECT GAS SCALE J since T A = 1/8. Thus to determine an unknown temperature the corresponding pressure *, is observed on the thermometer and he temperature 1\ ^calculated either from the above relation or graphical y. In an actual measurement corrections have to be applied for the following : — (a) The gas in the 'dead space' is not raised to the ^.nperatiire of the bulb. The Mead space' consists of the space }»*" <^f tube and in the manometer between the mercury level and the steel- piece H. Its initial and final temperatures are also different. , Increase in the volume of the bulb C with rise of temperature. (c) Change in volume of the bulb due to changes in internal 3S "d)' Changes intensity of mercury on account of temperature changes, For a discussion of these, the authors' book ',4 Treatise on Heat may be consulted. THERMODYNAMIC OR ABSOLUTE SCALE OF TEMPERATURE 7 Perfect Gas Scale.— The formulae developed for the gas thermometer assume that the gas, in question, accurately obeys the E?as taws but experiments show that no real gas does so exact!), ine coefficient of expansion « at constant pressure is not exactly the same for veal eases as may be seen from any book of physical constants {see Rave and Laby : 'PIMical Constants'). Further the two coefficients * Fare not exactly equal, and also varies with the initial pressure Thus, different gases would furnish different scales of temperature ij the thermometer is calibrated as indicated above, and the selection of anv particular gas will be arbitrary and will give an arbitrary scale of temperature. To avoid this arbitrariness we must reduce our ovations to that state of the gas.in which the perfect gas* equation i, satisfied. We shall now indicate the mediodsf of reducing the observations on real gases to the perfect gas state. This can be done when we have knowledge of the deviation of gases from Boyle s law. 1 he calculations are rather complicated and will not be given here. It is enough to point out that the equation for any real gas can be written in die form PV. z=-.RT + Bp +\Cp* + D / ,a + ' where! B, C, Z>...aie constants which go on decreasing rapidly. Thus ii is evident (hat at infinitely low pressure (p-*0) all gases will obey Boyle's law accurately and this conclusion is borne out by expert * A perfect gas is defined as one which will obey Boyle's law and Joule's law (Chap. II. sec. 22) rigorously. t For fuller information sec A Treatise ott Heat by the Authors. J Further see Chap. IV, Section 1. 10 II!' I!. MOM IMR1 [t KAP, rnent;. ation. Now it is mentally found that the tempe- rature scales, obtained by using the different gases and extrapolating observations to zero pressure, art: actually identical lor all the This is the perfect gas scale. The coefficient of expansion « for a perl: an be calculated in this manner from the data given by Heuse and Otto. The mean of several results gives a = 0.00 for a perfect gas. Thus T the melting point of ice on the perfect gas scale— l/« = 273.10°K, Considering all the available data Birge adopts the value 273.16^0.01. We shall use the value £7:: or 273. pending upon the degree of accuracy required. Aunin i ;j accurate experimental data on oxygen we have Lim ( 22.41-1 litre X atmospl Hence fl = Lim (pV)„ 22.414 x 10 a X 76 X 13.595 X 081 ., 97 * Tfi ergi/degree 273.16 = 8.314 X I(|T : 1.186 X ' 'I' 11 ' 1 ' III), hence R=l , per degree. This is the value of the gas constant. For real alue of the quantity p G f',,/7',, differs only slightly From this, v point out i alt arrived at in Chaj tvn how Lord Kelvin, of heat- scale independent of the property scale of temperature the thermodynamic scale. Further it is shown there i uite identical with the perfect gas scale. We thus see that th< . scale which was hitherto shown to depend on the properti; is now becomes independent of the properties, any particular substance. Hence it is called absolute scale standard scale adopted in scientific work. r method of obtaining the correction to be applied to the real gas scale consists in performing the Joule-Thomson experiment (Chap. VI). But unfortunately the existii g data on Joule-Thomson effect are no tent to enable us to apply this method and the method given above is almost universally employed. S. Standardization of Secondary Thermometer*. — Gas the^ntome- • are very cumbersome to use and require several corrcctio Hence in labor a to are replaced b; secondary standards, such as the resistance thermometer, the thermocouple, etc., which have been carefully standardized by comparison with a standard gas thermo- er in standardizing laboratories like the National Physical Tab. or the Bureau, of Standards in Wa RE BATHS I I pur] imparison baths may be constructed, each suitable foi particular range. Between 0° and* 100°C a water bath, between 80° and 250°C an oil bath, between 250° and 600 q C a mixture of potas- sium nitrate and sodium nitrate., and above that an electrical heater is generally employed, The secondary thermometers may also be standardized by means of a series of easily reproducible fixed points whose temperatures have been accurately determined. A table ot standard temperatures is given below (Table I.) . The values are generally those adopted by the r enth General Conference of Weights and Measures represi thirty-one natrons which was held in October 1027, but some amend- ment-, made by the Ninth General Conference in 1948 have also been incorporated.* 9, Fixed Temperature Baths. — It is frequently convenient to calibrate the secondary thermometers by means of the fixed-point scale given in Table 1. The ice-point may be most conveniently obtained by dipping the thermometer in pure melting- ice contained in a dewar flask. This is a double-walled glass or metal vessel whose sides are silvered.-]* For the steam-point the hypsometer indicated in Fig. , r >, p. 12 is employed. The diagram explains itself. C is the v I, — Standard Temperatures. Temper?' i Substance Temperature Substance Centigrade tvti nade -252.780° E. P. of Hydrogen 419*5 F. p. of : -195 B. P. 444 -vi B. P. of Sulphur - 182 :. P. of Oxygen 630 -5 F. P. o : Anil :i - 7S-5 Sublimation of CO» 9m -8 M. P g Ivor - 38-87 F. P. of Mercury 1063 M. P. 0-00 :■,:. p. of ke F. P. of Coppi + 32-38 Transition temperature 1453 U. P. ol Nickel of Na.SOaOH.0 1553 F. P. of Palladium ioo-orjo B. P. of Water 1769 F. P. of Platinum 218-0 B, I ■ > hLbalene 2443 F, P. of Iridium 231-9 R P. of Tin 2620±10 M. P. of Molyb- 305-9 B. P. of Bcnzophenone denum 320-9 F, P. of Cadmium 3380±50 M. P of 327-3 F. P. of Lead 3500+50 -M. P. of Carbon *See "The International Temperature Scale of 1948", National Laboratory, Teddington (1949), tFor a complete description see Chap. VI. 32 THERMO M KTRY [chap, condenser employed lo present the water from being lost by evapora- tion, M is the manometer and T the thermometer. The path of steam is indicated by arrows. The boiling point of water at the pressure p (in mm. of mercury) h found to be given by the relation * = 100.000-1-S.67 X I0~ 2 (p - 760) - 2.3 >< 10-* (p - 760)2. For other fixed points a number o£ vapour baths in which sulphur, naphthalene, aniline etc. are used, serves the purpose, _ For determining the boiling point of sulphur Callcndar and Griffiths found that the standard Meyer-tube apparatus was verv suitable. It consists of a hard glass cylinder A of diameter 5 cm. and length about 25 cm. to which a spherical bulb B is attached at the bottom (Fig, 6) , The whole is surrounded by an asbestos chamber C. The thermometer T is fitted with an asbestos or aluminium cone Fig. 5, — Hypsometer. 6. — Sulphur-boiling apparatus D. This cone serves in Lwo ways : (1) it prevents the condensed sulphur from running down over the bulb and cooling it below the temperature of sulphur vapour ; (2) it prevents the bulb from directly radiating to the cooler parts of the tube. Sulphur is placed in the bulb and heated aver a flame. A side tube may be provided in the upper part of the chamber and serves to condense sulphur vapour. The boiling point of sulphur is given by the formula t = 444.60+9.09 X 1«~ 8 (P -760)- 4.8 X 10 •"(/>- 760)*. PLATINUM THERMOMK'iKKS 13 C|P c Baths for naphthalene and aniline may be constructed by slightly modifying the above apparatus. RESISTANCE THERMOMETRY 10. Platinum Thermometers.— The necessity of , ndary standards has been clearly indicated above. :> tvpes of such instruments based on two electrical properties of matter will be described in this chapter. These properties are— (1) variation of electrical resis- tance of metals with temperature; (2) variation of thermal electro-motive force with temperature. First let us consider the former. Sir William Siemens was the first to construct ermometer in 1871 based on this principle but^he con- structional details were unsatisfactory. Later improve- ments by Callendar* and Griffiths have given the instru- ment its modem form. Fig. 7 represents an hermetic- ally-sealed thermometer designed by Dr. E. H. Griffiths for laboratory work of high precision. Pure platinum ire free from silicon, carbon., tin and other impurities is : ted. It is doubled on itself to avoid induction effects and then wound on a thin plate of insulating mica m. The ends of this wire are attached to platinum leads which pass through holes in mica sheets closely fitting the upper part of the tube, and the other ends of these ads ire joined to terminals P, P at the top of the instru- ct. The mica sheets give the best insulation and vent convection current of air up and down the tube, L e coil is sealed for, otherwise, moisture would deposit 'in the mica and break down the insulation. To cpm- iisate for the resistance of the leads, an exactly similar pair of leads, with their low ends joined togeth placed close to the platinum thermometer leads, and is , umected to terminals marked C, C, on the instruir, hese are called compensating leads and are joined in the third arm of the Whcatstone bridge as shown in r, 8 (p. lft). Then since the ratio arms are kept equal ;uid the compensating and platinum leads have equal resistance at all temperatures, it is the resistance of the inum coil alone which is determined. For work up to 700 °C copper leads may be used and the whole may l e : nclosed in a Lube of hard glass. But for high tem- Fjg 7^pi ati . peratures platinum leads must he used and the whole num. therm be enclosed in a tube of glazed porcelain. meter. The precision and reliability of modern resistance thermometers ate entirely due to the work of Callendar and Griffiths. They deter- * H. L. Callcndar (1863-1930) was educated at Cambridge and worked it; the I •, tidtsh Laboratory from ISSS to 1890 on resistance thermometry. His greatest are the development of the platinum resistance thermometer and the Investigation of the properties of steam. M MOMETKV MEASUREMENT Ol? RESISTANCE . latmum from 0° to 500*C and found '" ,L Itwas ™*y a ^n fay a parabolic formula of the = J2 «( 1 B*)* ■ . . . (7) C and 0»C and . , ,-,..9.J lO-» f = _$8 •- I0 : ■ it ,, i ' ' li « i -.. Srytosoh q B u U a £c equation to find | from the value ol [ e „ ave ' *™ tothl ! lario ,P™ ducednomeE 1,111 PWrt ";"• Thus we define the platn ans of the simple linear relation* Kinpen , t p by m ''-fc=S ,o ° US' SiS ill 00/ 'loo' • • *r-i? 100 100a -(- (100) 3 /? _ £(100)* r/ mi i ] a+ 100^ H 100/ Thus S in equation (9) is equal to x loo |> / - !0Q£* tie specimens emploved is nhnm i r, T t, •« I ' Ufa different from n . The exact relation may he fcduc, 4 finding the platinum temperature ^ lor the boiling point of sulphur temperature is known, and then substituting in ( Kir i tor a specimen, use my unkno ature. The the thermometer at the unkno temperature is found out and ; ( tf ) h ** determined Froi this, using the value of 8, the true temperature i can he determined will jip of (9). Il will be ol that the correction i giving the value of t-t t ins the unknown temperature t. voik the value ol t f may be substituted for t on the right-hand (9) . Tor accurate work, however, the procedure is as follows : — Tin hand side of (9) is calculated for different assumed values ol I able is constructed giving the value of this correction t< I different values of tp. With the help of this table the true tem- perature i corresponding to the experimentally determined platinum tempera i in r t p is found. It was shown later by Heyoock and Neville, and Waidner and Bin i if die platinum thermometer is standardized at Q", 100° boiling point o( sulphur the parabolic formula (9) gives true L! Ear as 630 °C We shall illustrate the method by, 'a numerical example. Let the resistance ol a given platinum tliermometer at 0°, 100° and i 4.6) be 2*56, 8. 56 and 6.78 ohms resj equired to calculate the true temperature when the resis ' E the. thei mometer is 5.56 ohi U-- 5 ,' 5 . e X 100 = 3GG°C. t, "~ 3-5G-2-56 for the boiling point of sulphur _ 6-78-. " 3-56- 444-6-422 =S 7,- * 100 ^- 422°C. 1 - 1 1 id / 100 hence & = 1.4. From (9) we get / = 300, (-*,== S.I-; i p = 291.2°, t — 320, t-t p = 10.4 hence f. = 309.6°. ►rrection t-t p for t f = 291.2° is 8.8 and for 309.6 it is 10.4. : 300 it is B.B + ~=^-X (300-291.2) = 9,6. rherefore the true temperature t is equal to 309.6 11. Measurement of Resistance. — The determination or tempera- . this thermometei Involves the accurate measurement ol the ace of the platinum wire. Various special types of resistance re used lor this purpose. In order to compensate for the ice ol the leads, a bridge with equal ratio arms i- used. Fui the brii g< should be capable of measuring cbaxi e to a ree of accuracy & fundamental interval is > and measurement of temperature to hundedths of a tl requi stance measurement to one ten-thousandth of an ohm. 16 THitRMOM I-TRY CHAP. Fig'. 8j— Calliflniaratid Griffiths Bridge. adjusted for no deflection of the galvanometer G. Then The Callendar and Griffiths bridge* is quite suitable for this pur- pose. Fig, 8 indicates r.he connections. Q and Si the ratio anas, arc kept equal by the makers of the instrument. R consists of a set of resistances of 1, 2, 4, 8,' 16, 32, 64 units. The usual plug contacts are here replaced by mercury cup contacts. L x and L-. axe two paral- lel wires of the same material which can be connected to each other by the contact-maker K, This arrangement is adopted in order to eliminate thermo-elec.tr o- motive forces. P represents the thermometer and C the compen- sating leads. The resistance r acts as a shunt and makes the resistance of the wire exactly in the desired ratio. The bridge is Suppose the balance point is obtained with the key K at a distance x from the centre of the wire and die entire length of the wire is 2a. or P =r.R — 2x P . where p is the resistance per unit length of the wire. It is found it to select the wire L 3 and che shunt r in such a mannei that 1 cm. o£ the bridge wire has a resistance of 1/200 ohm and the total of the wire is 20 cm., while the smallest resistance in R d 1 has a r: >. d stance of tfie whole i.e.,, 20/200 — 0.1 ohm. Thus if the fundamental interval of the it'Kc thermometer is 1 ohm the temperature can be determined C provided the balai at is determined correct to Oil mm. For accurate work., however, various precautions are neces- which are given belowj- ; — The current, flowing through the bridge heats the bridge coils and changes their resistance. The change in tem- perature may be observed on a thermometer and the corresponding change in resistance calculated. The correc- tion can then be easily applied. Or the bridge ni;i placed in a thermostat. The thermometer coil has to be very thin (0.15 mm. dia- :,Y) since it must have a large resistance and hence the heating effect is considerable. From Calendar's observations the heating effect for a current of .01 ampere is 0.016° at 0°C and 0.017° at 100 C C. According to him the best, procedure is to pass the same current through * For further details see Flint and W'or-v.o-j 'I'mciiaii I'kys'u-.-;'. i For full details see Methods of Measuring Temperature by E, Griffiths, (Chap. 3.) (2) B i 1-: A Sl.I REMENT OF RESIST A - 17 the thermometer at all temperatures when the heating effect remains approximately constant. (#) The bridge centre must be determined and the bridge wire calibrated. (4) Due to temperature gradient along the conducting leads and the junctions thermo-electromotive forces are developed in the circuit whose magnitude may be found by closing the galvanometer circuit when the battery circuit is kept open. To eliminate these the galvanometer circuit should be permanently closed and balance obtained for reversals of the battery current. If induction effects are perceptible when the battery * circuit is made or broken a thermo- electric key should be employed. This key first breaks the galvanometer circuit, then 'makes the battery and the galvanometer circuits in succession. (5) The external leads connecting the terminals PP, CO to the bridge should be exactly similar and similarly placed. 12. As already mentioned the platinum thermometer is standar- dized by measuring the resistance at the melting point of ice, boiling points of water and sulphur. The last gives S and hence t can be determined from any subsequent determination of t p . Eqn. (9) how- ever does not hold much above 630°C and therefore every such thermo- meter is provided with a calibration curve drawn by an actual com- parison with a standard gas thermometer in standardizing laboratories. The temperature can be directly read from this tempera We-resistim ,e curve, The great advantage of platinum thermometers lies in their wide- range (~20Q°C to 121KPC). If carefully prepared, their readings are reliable to 0.01° up to 500°C and to 0.1° up to 1200°G but generally it is not desirable to use them above 1000°C owing to the dan- : contamination toy the insulating materials. They are free from changes of zero for the wire when pure and well-annealed has always the same resistance at the same temperature. They are very con- venient for ordinary use n.nd, when onee .standardized by comparison ■ ith a gas thermometer, they serve as reliable standards. They are employed to measure' small differences of tempera tit re very ately, sometimes even to one ten-thousandth of a degree. There re, however, some drawbacks also. The resistance thermometer has a J: tge thermal capacity and the covering sheath has a low thermal Conductivity and therefore the thermometer does not quickly attain ilt:- temperature of the bath, in which it is immersed. Further some i ! Lost in balancing the bridge. For these reasons the resistance lometer is useless for measuring rapidly-changing temperatures. urther impurities in the platinum do not obey the same resistanre- I mperature law as the pure metal. I 2 gives the variation in resistance of a platinum therm o- metei over a wide range. It is taken from Henning's Temj . The value of the quantity R = R t /R is given for various o i;-, THERMOMETRY [chap. temperatures where H„ R are the resistances at tempera tine-:. m! ii ( ! respectively. Table 2,— Values of R—R t /R^ Ten:- . R Temp, R Ten J? D C 021607 °C -m +40 1.15796 280 2.06661 -180 0.25927 60 1.23624 300 2.13931 -160 0.34505 80 1,31406 320 2.21154 -140 0.42986 100 1.39141 340 2J28330 -120 0.51347 120 146830 360 2.35460 -100 O.S9i 140 1.54471 2.42543 - 80 0.67814 160 1.621 400 2.49S80 - 60 0.7594$ 180 1.69616 420 .570 - 40 200 1.77118 440 2,63513 -J 20 0,92033 220 1.85474 •li-:i 2.70410 1.00000 240 1.91983 480 2. 77261 20 1.07921 260 1.99345 THERMO-ELECTRIC THERMOMETRY 13. Thermo-Couples. — Let us now return to the second electri- asurement. Starting from ■ ., in 1821 nm ■• were made to ometer based on this principle, Eor instani Pouille( and Regnault. At present thermo-electric attained a degree of precision inferior only to rice thermometry below 1000°C, but foi atrires exceed- ::ic it [g the onl) sensit: I convenient electrical method at our disposal. A thermo-electric thermometer installation consists of the follow- ing parts :— ;]; The two elements constituting the thermo-couple. ' The electrical insulation of these wires and the protecting; tubes. (3) Millivoltmeter or potentiometer for measui in- the thermo- electromoi h e force, (4) Arrangement for controlling the cold-junction temperature. The choice of the elements constituting the couple is determined tperature to which the couple is to be heated and .the e.m.f. developed. For low temperatures up to 800°C couples of base mewls such as iron-eon sf a ntnn and copper-constantan are satisfactory, as they lop a large e.m.f. of about. 40 to fif) microvolts per degree. For H atures these base metals cannot be used as thev get ized and melt. Nickel-iron couple may be used up to 600° while nickel-nichrome and chromel-alume) thermo-couples can be used up '-] THERMO-GOUP LES 19 to 1000° but above that platinum and an alloy of platinum with iridium or rhodium must be used. Le Chatetier in 1886 introduced the couple consisting of pure platinum and an alloy of 90 per cent Pi and 10 per cent Rh which is no employed for scientific work. The e.m.f. developed by these noble metals is, however, much less. The two elements are taken in die form of a wire aud one end of both is welded together electrically or in an ox y hydrogen flame. This end « (Fig, 9) forms the hot junction. The portions of the wires near the hot junction are insulated with capillaries of fire- clay (or hard glass for lower temperatures) and are threaded through mica discs enclosed in outer protect- ing tube of porcelain, quartz or hard glass, depending upon the temperature for which it is meant. The protecting tube prevents the junction from contamina- tion but necessarily introduces a lag. For rough use this may be further enclosed in a steel sheath (shown black in the figure) . Where there is no risk of conta- mination, the mica discs and the protecting tubes can be dispensed with. The wires are connected to terminals d ana! C s on the instrument. To these terminals are connected flexible compensating leads leading to the cold junction ' (Fig. 10a) . These leads are usually of the same material as the elements of the couple itself. Thus the cold junction is transferred to a convenient 1 1 hi mt place where a constant temperature, say Q C, can be maintained. Usually the compensating leads Hi; marked so that there is no difficulty in connecting jo the proper terminals. There are two ways of making the connections which are indicated in Figs. 10 (b) and (c) . The dia- grams explain themselves. The cold junction is immersed in ice at 0°C. As a recording instrument r a millivoltmeter or a potentiometer is employed. -a Fig. 9 — Thermo- couple. i i couple Cold JtUlCtiun Hat Illl'i til I- Cotnpeasating extension Hot junction junction junction W (6) Fig, 10.— Measuring with a tincnno-couple. :v: 14. To Unci the temperature of the hot junction we must tire the e.m.f, &< bei « een thi enda ' : me o ►pper leads. 20 THERMOMETRY [CHAP. I li : s ran be done by means of a high resistance miilivoltmeter which mav be graduated to mperatures directly and die temperature thus obtained can be relied upon to about ±$°C. For attaining higher accuracy Fig. H.— Illustration of the principle of potentiometer. a potentiometer must be used. This arrangement essentially consists of a number of resistance coils A (Fig. 11) placed in series with a long wire resistance r stretched along a scale. A current from the battery E flows through these re- sistances and its sti: is so adjusted by varying R that the potential differ- ence across a fixed ■ ance X balances against the c.mX of a standard cadmium cell C {L0183 volts) . The e.m.f. developed by the thermo-couple Th is balanced as indicated. The potentiometer can be made direct reading bv keeping K=z 10 LP ohms. Thus there is a fall of 1 volt pe ohms and by constructing; the smallest resistance coil of fl.l ohm mce and the wire r also of the same resistance, the total e.m.f, : the wire will be 1 m.v. If the wire is divided into 100 d.iv and in addition lias a sliding vernier having TO divisions the readings )e taken correct to 1 microvolt. Various types of potentiometers based on this principle have been devised specially lor this purpose* With these insiniments the e,m.f. can be measured accurately to 1 microvolt which corresponds !,, r a pt— Pt-Uh couple. For a copper-constantan couple this corresponds to about 1/40 degree. With a sensitive arrangement it. is possible to measure to 0,1 microvolt when the sensitiveness is increased about ten times. For accurate work the cold junction must be maintained at 0°C otherwise corrections! will be required in that respect. In order to deduce the temperature from an experimental deter- mination or the e.m.f. a calibration curve is generally^ supplied with the instrument. This gives the temperature corresponding to different electromotive forces developed and has been drawn by the makers by an actual comparison with a standard thermo-couple Throughout the range. If it is required to calibrate a thermo-couple in the absence of a standard one, the fixed points (Sec, 8) must be utilised. The e.m.f. at those points is measured and an empirical interpolation formula employed in order to give the e.m.f, corresponding to the * A description of these will be found in Methods of Measuring Temperature, by E, Griffiths. f See Ezcr Griffiths, Methods of Measuring Temperature, (1947), p. 74. THERMO-COUl'LhS 21 rmediate temperatures. For a Pt— Pt-Rh couple three different equ: must be used for the different ranges. Thus ; am 0° to 400*0, E = At + &(l- O , 300° to 1200 C C, E = - A* + & + &?, 1100° to 1750<>C, E = -A" + B"t -f- C"l* where A, B, C are constants whose values are empirically determined. Thermo-couples are frequently employed for laboratory work since v are cheap and can be easily constructed. They can be used for i measurement of rapidly-changing temperatures since the thermal -city of the junction is small ami hence the thermometer has . ctieally no lag. Another advantage in the use of thermo-couples is they' measure the temperature at a point—the point at which the two metals make electrical contact. Its chief disadvantage lies the fact that there is no theoretical formula which can be extra- polated over a wide range and consequently every thermo-couple requires separate calibration. The useful range of thermo-electric thermometers is about - 200 & to 1600°C. Readings are reliable only, when the composition of the i ile does not change even slightly. In actual practice frequent calibration is necessary. The following are the chief sources of error in thermo-electric i mometry : — (1) Parasitic electromotive forces developed in the circuit. They are due to (a) Peltier effect or e.m.f, developed due to heating of . t.i.ort of dissimilar metals at points of the circuit other than the i and the cold junctions. This occurs often in the measuring >aratus ; (b) Becqucrel effect or e.m.f, generated due to inhomo- geneities in a single wire; this occurs mainly in the thermo-couple wires. The e.m.L measured is a sum of these quantities and the Peltier e.m.f* at the two junctions and the Thomson e.m.f, along homogeneous wires of the thermo-couple with ends at the two tempe- ratures. The undesirable effects mentioned in (a) and (b) must be dilated by the use of materials and methods free from these effects nee they are not taken into account in any thermo-electric formula?. (2) Leakage from the light mains or furnace circuit. If leakage s through the potentiometer their presence can be detected 1 1', short-circuiting the thermo-couple when the galvanometer continues to be deflected. I) Cold-junction correction if it is not kept at (PC. For the methods of minimising or eliminating these errors the is referred to Measurement of High T wres by Le Chatelier and Burgess. Table 3 compiled from various sources gives the thermo-electric J. for various couples in common use. The cold junction is • i, -lined at 0°G* and the hot junction at t°Q. The e.m.f. of the '•no-couple AB being positive means that the current flows from \ to P. at the cold junction. 22 THERMOMETRY [CHAP. Table 3. — E.m.f. in Millivolts for some thermo-couples. Temp. 90Pt-10Rh .'Ni Ag/Pl Fe/cons- Cu/cons- fC against Ft - tarrtan tantan 200 -S.27 --. -100 ■ 3.349 - 80 — 1.68 -0.30 r;.co 0.00 ;:i::i 0.00 0.000 + 1 +0.643 +5.18 +0.72 I 5.40 +4.276 200 1.436 -.96 +1.73 10.99 9.285 m 2.315 752 2.96 16.56 14:' 400 3. 250 9.83 4.47 500 4,219 12.04 6.26 27,58 l.llii 5.222 1450 8.25 700 S.260 17.30 39.30 800 7.330 20.73 13,17 45.72 000 8.43 24.19 15.99 9.569 . US 14.312 1600 16,674 17 Ml 15. in methods of measuring temperature utilise the radia- ; to he measured. will be discussed in detail later (sec Chap. XI). "J I for measuring temperatures from about lOOO^C to any upper limit. 16. Certain other methods of measuring temperature utfljse any one of the following properties of matter : — (1) Expansion of a bar of metal. (2) Changes in vapour density with rise of temperature. (3) Variation of refractive index oF a gas with temperature accordance with Gladstone and Dale's law. ": Calorimetric methods based on the measurement of quantity oi h : Change of vapour pressure with temperature. 17. Low Temperature Thermometry.* — The standard thermo- meter in this range is the constant volume hydrogen or helium thermo- meter. The difficulty in this case is that gases liquefy and even lify at these low temperatures. Prof. Dewar, however, showed that the boiling point of hydrogen as indicated by the liydrog thermometer was -253.0 o C and -253.4°C, while a helium thermc meter registered -253.7°C and -252.1°C, Similarly he compared * For pyromelry see Chap. XT. LOW TEMVl'UATURE THERMOMETRY 23 thermometers of other gases. These experiments evidently led to the conclusion that a gas could be relied upon almost to its boiling point. Thus helium furnishes the scale down to its boiling point (4.2° K.) . The corrections necessary to convert this scale to the thermodynamic scale may be obtained and have been given by Onnes and Gath. For temperatures below 4.2°K we must use the helium gas thermometer with die pressure well below the vapour pressure of liquid helium at the temperature to be measured, so that the gas will not liquefy at that temperature. With this device the helium gas thermometer gives us the thermodynamic scale down to 1°K. Now we shall consider the secondary standards. Mercury freezes at -38.87 and alcohol at -1]1.8'"C and hence these thermometers cannot be used below the respective tempera- tures. A special liquid thermometer containing fractionally distilled petroleum ether can be used down to - 190*0, But for all accurate work, however, resistance thermometers are employed. It is absolutely essential that the substance of which the thermometer is made is perfectly pure. Pure metals show a regular rise in resistance with decrease of temperature. Dewar and Fleming found that the presence of the slightest trace of impurity in a metal is sufficient to produce a considerable increase in resistance at these low temperatures. It is, therefore, difficult to trust the purity of any specimen for very low temperatures without actual comparison. Henning found from a detailed investigation that the parabolic formula did not hold below — 40 D C. Van Dusen proposed the formula where e is a constant and 8 has already been defined on p. 14. The constants, R lf! K im and 8 are determined by calibration at 0*0, mil c; and the boiling point of sulphur as explained previously, and the constant e is then determined by calibration at the boiling point of oxygen (- I82.97°Cj . Van Dusen s formula has been found to hold atisfactorily from 0°C to -190°C, die error nowhere being greater ban ri=0,05 a . For temperatures lower than - 190°C, the platinum thermometer was used by Henning and Otto, and can be used with advantage up to 20°K. There is, however, no satisfactory, formula for calculating the temperature from the observed resistance and a calibration curve has to be used. Sometimes lead and gold thermometers are also employed. Onnes has used lead down to -259°C and Nernsl lias, given a method dculating these temperatures. Below —250*0 resistance ther- mometers of cons tan tan and phosphor-bronze have been employed, the latter being much more sensitive. For low temperatures copper-cons tan tan and iron-cons tantan couples are very sensitive as they develop a large e.m.f. They can be used down to - 255 B C. They must be calibrated by direct' com- Eson with a gas thermometer. ^4 THERMOMETRY In order to measure temperatures below the temperature ol helium (-268°C) the vapour-pressure thermometer o£ helium can be employed, Its use is based on the well-known fact that the vapour pressure of a liquid varies uniquely with the temperature. Thus the method consists in measuring the vapour pressure of a liquid at the required temperature by means of an apparatus similar to that shown in Tig. 4, Chap. V, and obtaining the corresponding temperature by means of a calibration curve or a theoretical formula. The helium gas thermometer and the vapour-pressure thermometer have been used down to about 0.75 C K. For measuring still lower temperatures the paramagnetic susceptibility of salts is utilised. 18* International Temperature Seal*.— We have seen that the thermodynamic centigrade scale is the standard scale of temperature d is given by the helium gas thermometer, but gas thermomel involves many experimental difficulties. On account of these difficulties in the practical realisation of the thermodynamic scale the Inter- M iiovial Commitee in 1927 found it expedient to adopt a practical scale known as the International Temperature Scale. This scale agrees with the thermodynamic scale as closely as our present knowledge permits and is at the same time designed to be easily and. accurately reproducible. It is based upon a number of reproducible fixed points to which numerical values have been assigned and the intermediate temperatures have been defined by agreement as the values given by trie following thermometers according to the scheme given below ; — (Pi 9 1 and and platinum resistance neter calibrated at D , I00°C and the boiling point sulphur. : From -190 platinum resistance thermometer which gives temperature by means of the formula Et = R (I -f- <xt + fit 2 + y (« - 100) **}, the four constants being determined by calibration at ice, steam, sulphur and oxygen points. It, will be seen that i.iv- formula is equivalent to (10) where e s= (I00) 2 87//J. (3) From 661>°C to I0b"S°C.— The platinum Pt-Rh thermocouple where temperature is defined by E = a -f bt 4- cF r and the three constants are determined by calibration at the dug point of antimony and at the silver and gold points. (4) Above 1063°C— An optical pyrometer (see Chap. XI) cali- brated at the gold point (106S D C) . it should be emphasized that the International Scale does not replace the thermodynamic scale ; it merely serves to represent it in a practical ter with sufficient, accuracy for most purposes. >•] 0«K- lOO'K- 273-K- IflO'C- 200 300 m SCO 6Q0 TOD — SOD — : :o :;. :: CHART OF FIXED POINTS Ch art of Fixed Po ints -iS-iOLi'. B. P. of Helium- -283.-7E 1 B, V. of HydrDgeu -LS5'6J B. P. of Nitrogen ^Ug-S«a B- P. of Oxygen - 78-4E3 Sublimation of CO* - ss-632 F. P. of Ifenury D'OU Water freaes • 9S'3S4 TroBsSttonofNa^lOHjO. ■ lOft-Ott B, P. of Water 2?) 419 '43 4,14 '60 SITS* B. P. of. Naphthalene 231*86 P.P. of Tin 305-90 B. P. of Bensoptienmic 320-9 F . P. of Cadmium 1 r - B. P. of Sulphur ' .•■ .:, ' . . 523*C. Draper point iBed Just Visible SKI'S F . P. of Antimony 600' I F. E of Alurnjniym m-9, F, P. of Silver Dull red Cherry red Bright □harry 1 & ■-; -' 26 i -.-"-I i I'-I.M liO'J !':'•' 2400 SWO THFRMOMKTKY Chart of Fixed Points (Contd}* [chap. M. P. of Gold p.P.ofCopJn:r ■1*53 F.P. ofNkNri ■15S2 F.P of Palladium 176& F. P. of Platinum F.P. of Iridium P. pf Molybdenum MOO H 40MT- ■ 3850 M, P, of Timgaten Dun Oxaage Bjiebt Orange Whit? Dasslme 1 I M ILLUSTRATION OF PRINCIPLES OF THERMOMETRY 19. Illustration of the Principles of Thermometry, Absolute or Thermodynamic Scale Gas Thermometers (Primary standards) 27 Constant Volume Constant Pressure Gravimetric Thermometers (Compensated Air thermometer) 'Secondary Thermometers r — r T" ~i Expansion "Resistance Thermoelectric Radiation Thermometers Thermometers Thermometers Pyrometers Bo oka R ecom men (led 1. Burgess and T.e Chatelier, The Measurement of High Temperatures. 2. Ezer Griffiths, Methods of Measuring Temperature, (Griffin, 1947). ■. Hcrniing, Temperaturmessung. 4, A Dictionary of Applied Physics (Glazebrook), Vol. J, Article on Thermometry. 5, J. A, Hall, Fundamentals of Thermometry, Institute of Physics, London (1953). 6, J. A. Hall. Practical Thermometry, Institute of Physics, London (3953). Other References-. Temperature, its Measurement and Control in Science and Industry (1941) , published by Reinhold Publishing Corpo- ration,, New York. METHOD OK MIXTURES 24 CHAPTER II CAL0R1METRY 1. Quantity of Heat. — It is a matter of common experience that when a hot body is placed in contact with a cold one, the former becomes colder and the latter warmer ; we say that a certain quantity . of heat has passed from the hot body to die cold one. But a sim| experiment shows that when different, bodies are raised to the same temperature and then allowed to exchange heat with a cold body the final temperature is different. If we take equal quantities of water in three different vessels at the same temperature and plunge equal masses of aluminium, lead and copper previously heated to JOO'C into diese vessels, one in each, the equilibrium temperature is highest for aluminium and least for lead. This indicates that, of these three metals, aluminium can yield the largest quantity of heat and lead the le For measuring quantities ol: heat we require a ' unit/ The quantity of heat required to raise the temperature of 1 gram of water through 1°C is called the 'calorie' which is also the thermal unit tot measuring quantities of heat. The 15°C calorie is defined as the quantity of heat, which would raise the temperature of oi gram of C to 15.5°C and has been re ided hv the lied Physics (1934) for andard. In Britain the British thermal unit tly employed* which represents the qua heat required to raise 1 lb. of water through l°I\ 1 of any substance is defined as the number of calories . u of the substance through 1°C. Tin's is peaking not the same at all temperatures. Thus if a quantity . the temperature of m grams of a substance from b to 8% s f the mean specific heat of the substance, is given by | m {B'~9) ] ; while if a quantity dQ raises the temperature by d&* tin specific heat at the temperature $ is —^ , The thermal capacity or water equivalent of a particular body he product, of its mass and specific heat. 2. Methods in Calorimetry.* — 'The following are the chief methods iloyed in Calorimetryf : — (1) Method of Mixtures. (2) Method of Cooling. * Sometimes the lb. calorie or cen%rade heat unit ("C. H. U.) or centigrade thermal unit (C. Th, U.) is also used which represents the quantity of hi ed bo raise 1 3b. of water through 1'C. good account of these method * js given in Glazebroafe, A Dictionary of Physics, Vol. I, article on "Catorimetry". (») Methods based on Change of State or Latent Heat Calorimetry, (■1) Electrical Methods. In die following pages we shall discuss these methods one by der each of these we shall consider the various forms of mental arrangement that have been adopted. Solids and liquids will be considered first while gases will be taken up later in the iter. L METHOD OF MIXTURES 3. Theory of the Method.- — Regnault* about the year 1840 made ,i careful study of the Method of Mixtures, and by 'care and skill icd results of the highest accuracy. The principle of the m. is to impart the quantity of heat to be measured to a certain mass of water contained in a vessel of known thermal capacity and to tire the rise of temperature produced. Thus, if a substance of ,72-,., specific heat s t and initial temperature 6 Xf be plunged into rams of water at temperature 0g* and if W be the thermal capa- of the calorimeter, the final temperature of the mixture, we 1 1 i : . i. • , by equating the heat lost by the ice to the heat gained by the water and calorimeter, lis gives the specific heat of the substance. Various correc- r, necessary for heat is lost by the system by luction, convection and radiation. Thus lor 8 we must put ' e /S& is* the correction. A. Radiation Correction, — In most experiments on calorir the calculation of this loss of heat due to radiation is important. ITic radiation correction may be accur- al Hy calculated with the help of ton's Law of Cooling (Chap. XI) & h states that for small differences of temperature the heat loss due to iation is proportional to the tempera- difference between the calorimeter i iul the surroundings. To illustrate its application kit AB (Fig. 1) denote the rved rise of temperature during an ent, RC the observed cool: i, I of it. We have to calculate rue rise in temperature*. Divide the abscissa into n equal intervals 5.',. :.i....St B by means of ordinate* P 1 M ii P 2 M 2 , . . .P„M S such that * Henri Victor Regnault (1810-1878), born at Aix-Ja-Chapclle, had to support himself while young, He joined the Ecole Poly technique in Paris and later on in [840 he was appointed Professor in that PoJyteehnique. He r.id many classic . on heat. AM.MW W« ■t-f fl + D Fig. 1. — T] lustration of Radiation CoiTU'/li: 'I. 30 CALORIMETRY CHAP, the small portions AI^ PJ* . .. P^P., may be treated as straight ines. Let us measure temperatures from th< ie temperature or the surroundings. If 8 X , 2 . . . denote the mean temperatures during these intervals, B t ', 0/ . , . the temperatures at the ends of these intervals represented by P^, P 2 M 2 . . „ then the temperature diminution due to radiation in the interval gr^ is kBtfh. If ${, 0*, . . . denote the temperatures at die ends of these intervals had there been no loss due to radiation, then 0-"^ = 6, t "= #,' -- kB x U x a "= 0/ -f MjBtj, -|- ke.,bt 2 . + Af^&i -!- 8 a/ 8 -f- .. .. dji„] + £(area of the curve ABM„A) = V + k 'j: e *//. (i) We can thus correct any temperature 0/ AP.M.A and k. if we determine the area An alternative method is to plot the upper curve from the lower curve by increasing the ordinate M t P x to MiP/, M,P. to M«P,' etc where M^P/ = ^ HP,'=V. etc. The highest ordinate "on the curve (wo. D£) gives the true rise of temperature in the experiment corrected for radiation. r To determine h we have to observe the rale of cooling at anv temperature. The curve BC {Fig, I) is obtained experimentally for this purpose. From this ~ is calculated for any mean value of $. is known.* Another method called the adiabatic method is to eliminate the teat bj continuously adjusting the temperature of the bath enclosing me calorimeter to be always equal to the temperature of the caTori" ureter itself.* 5. Specific Heat of Solids. — For finding the specific heat of solids by this method the requisites are a calorimeter with an enclosure, a thermometer and a heater. For work at ordinary temperatures the calorimeter is made of thin copper, uickelplatcd and polished on the. outside., so as to reduce radiation losses, it is supported on pointed pieces of wood or by means of thread inside a larger double-walled vessel which has water maintained at a fixed temperature in the ,.;, t,A« fl rt nple b v fc ro V; ?h m ! thod sometimes adopted is to add to the observed rise halt the coolwyr observed at the hi R hest temperature m a time equal to the duration .of the experiment. This is based on the assumption that the average excess of temperature of the calorimeter over the surroundings may be taken In «Uw i tt i T^ S > hL T cc the co ° ]in « durin ff the experiment is half th cooling at the final temperature. * tt F Ja other methods £cc Glazebrook, A Dkiiottary of Applied Physics, Vol. 1 SPECIFIC HflAT OV LIQ1 '" ■ 31 D la space between the walls. The heater is a steap-jacket in ,, i substance is heated by steam without becoming wet. An oil bath i i be used. The transference and radiation errors must i by suitable mechanical devices as in Rcgnault's classical in ents. For high temperatures the solid substance is heated in an electri furnace. While in his work at high temperatures employed a furnace h .. in •■ .■ platinum coil wound on its surface. The substance is sup- inside the furnace in a loop of platinum wire up! is allowed to drop into the calorimeter by a i i mechanical device. Change in temperature is measured by a resistance thermometer. For work m tow temperatures the substance is cooled down in a quartz vacuum-vessel surrounded by liquid air before being dropped into the calorimeter. Awbery and E/er Griffiths have determined the irk heat of solids and molten liquids as well as i latent heat by using an improved apparatus i iid on the method - of mixtures. This is discussed in Chap. V. The use of water as calorimetric liquid has eral drawbacks. Its range is small and specific large so that the rise of temperature is small ; lier there is considerable risk of some water being 1 1 i,i by evaporation. For these reasons several workers have replaced it by a block of metal The copper block calorimeter devised by Nernst, Lindernann and Koref is exceedingly convenient for low temperatures. It consists of a " heavy copper block X (Pig. 2) .Hinted with Wood's metal to the inside of a Sewer flask D. It is essentially a calorimeter based on the method of mixtures in which copper replaces water as the standard substance. The heated substance is iped into die copper block through the glass-tube 11 and the change in temperature of the latter is read on thermo-couples T, T, whose one end is inside the copper block K and the other end in the block G. The copper block on account of its good conductivity keeps the temperature uniform. Jaeger and his co-workers have employed this method to determine the specific heat of W, Vt, (X Rh, Ir, etc., to about, lGOn^C with a high degree of accuracy. 6. Specific Heat of Liquids. — Specific heat of liquids which do not react chemically with water or any other substance or known specific heat may be obtained by direct mixture. For liquids which react in this way Rcgnault used a different form of apparatus. The liquid was not allowed to mix with water but was admitted when desired into a vessel immersed in water. The liquid was first heated •T- Ffc. 2. Copper Block Calorimeter, CAl.ORIMETRY [CHAP. The specific heat could be calculated and then forced into the vessel. as before.* Another cla*s ol experiments For measuring the specific heat, of liquids involves the expenditure af some mechanical energy and measurement of the consequent rise of temperature. To this class belong die classical experiments of Rowland for determining the ichamcal equivalent of heat. They will be described in detail in Chapter 11L 2. METHOD OF COOLING 7. This method, perfected by Dulong and Petit, is found to be most, convenient for liquids but unsuitable for solids owing to varia- tions of temperature within the latter. The method is based on the assumption that when a body cools in a given enclosure on account of radiation alone, die heat dQ emitted in the time dt is given bv die relation d(l=Affldt, . . . . ,(4) 1 depends upon the area and the radiating power of the Em f{9) is an unknown function of $, the excess of the temperature of tile body over that of the surroundings. I! :' ; produce* a cooling; of the body through -d$, we have dQ = -msdd, iote the mass and die specific heat of the body ng these two expressions for dQ we get - made =3 Af(&) dt. Or f .. m * a dO Or t __ ms f* 1 ii8 _ " * )&*Mr ' ' - ' - I 5 ) e * is the time the body takes in cooling from X D to a 2. Similarly for another substance to cool through the same" interval Ml LcMl'J J _lcl <X Lll] C ' A' )B s f(d)' ' ( 6 ) * i- ~: r i e * Che sutfa f e area and the radiating power- of the two bodies be rbx: same we have from (3) and (6) ms m's' t =- r- ■ ■ - (7) If masses m, m> of two liquids be contained successively in a- calorimeter o| thermal capacity W and the calorimeter Upended * For details see Preston, Theory of Hmt u MFTUOD OF MILTING ICE , inside a vessel kept at 0°C by immersion in melting; ice and then .a lions of the rate of cooling taken, we have W_ ■ t *o (S) II' one liquid is water (s — 1) , the specific heat of the other is thus determined from a knowledge of t, t', m t m'. The method is sometimes employed for determining the specific li it of liquids but is not capable of any great accuracy and is mainly of historical interest. 3. METHODS BASED ON CHANGE OF STATE 8. These methods may be subdivided Into two : namely the method of melting ice and the method of condensation of steam. These methods were of real advantage in the last century when accurate measurements of temperature were impossible, but with the recent development of accurate thermometers and electrical heaters they are now less in use, chiefly on account of their inherent defects. The second method is, however, very convenient for determining the specific heat of gases at constant volume and hence retains its importance. 9. Method of Melting Ice.— In this method the heat given out by a certain substance in cooling is imparted to ice and measured by the amount of ice thereby melted. Thus ir M .grams of a substance of speciQc heat a and initial temperature & are able to melt m grams of ice when placed in contact with the latter, the specific heat is given by the relation MsB — m.L, . . . . where L is the latent heat ol; fusion. The earliest forms of the appa- ratus as devised by Black, and by Lavoisier and Laplace were liable to cause considerable error. An improved form of the calorimeter was later devised by Eunsen* which will now be described. 10. Bunsen's Ice Calorimeter. — In this calorimeter, the water produced by the melting of ice is not drained off but is allowed to remain mixed with ice and the resulting change in volume is observed. The calorimeter is illustrated in Fig :<. p. 34. The test tube A is fused into the cylindrical gla^s bulb B which is provided with the glass stem C. B is nearly filled with boiled air-free water and the remain- ing spare and the stem is filled with mercury. The stem terminates an iron collar D containing mercury into which a graduated tillary tube E is pushed so that mercury stands at a certain • n in E. In conductio , an experiment a stream of alcohol, cooled by a ezing mixture, is first passed through the test-tube A until a cap ♦Robert Wffliebxi Bi 111-1899), born at Gottingea, studied at Gottingen, Paris, Bcriin and Vienna. He r of Chemistry at Bresfau and berg. His important researches arc on spectrum analysis Bunsen cell, '. i;- •:•••- gas tntni :r and ice calorimeter. 3 34 CALORIMKTRY [CHAP. of ice F is formed round it in B. The whole instrument is then kept immersed in pure ice at 0°C for several days till all the water in B is frozen. It is then ready, for use. To calibrate the scale on E, let a mass m of water at a tem- perature 0°C he poured into the test-tube. Some ice in B melts and the resulting contrac- tion of mercury, say n divisions on E, is observed. Then if a recession of mercury by 1 divi- sion corresponds to q calories of heat rn$ = nq ; or q = mQ/n .,■ (10) Next the substance un4er i vivc:, ligation, previously heated to a temperature &', is drooped into some water at 0° contained in the test-tube A. Then if M, Fig. 3^-Bui Calorimeter. s d enote the mass and the spe- tanoe respectively, and v? the observed reo oJ mercury thread in E, we have Mse' = n'q, or m q " Mnff (11) ill ilic heat of the substance. The specific heat, of rare metals which can be had in small quantities can be readily I by this method. The apparatus is, however, not capable of -rear accuracy. A fundamental objection to the use of the ice calorimeter rests on the fact that a given specimen of water can i: into ice of different densities, 11. Joly's Steam Calorimeter,— In the steam calorimeter devised by Prof, Jolv in 1886 the heat, necessary to raise die temperature of a substance from the ordinary tempera Lure to the temperature of steam is measured by the amount of vapour condensed into water at the same temperature. It consists of a thin metal enclosure A (Fig. 4) , double-walled and covered with cloth, which is placed lath a sensitive balance. One pan of the balance is removed and from this end of the beam hangs freely a wire w supporting a platinum pan inside the enclosure- The substance whose specific heat is required is placed on this pan and weights added on the other pan till balance is attained. The temperature of the enclosure is observed by means of a thermometer inserted into the chamber, and in the meantime tL] jqly's steam calorimeter 35 steam is prepared in the boiler. It is then admitted suddenly into the chamber through the wide opening O at the top and can escape Fig. 4. — Job's Steam Calorimeter, through the narrow exit-tube t at the bottom. Steam condenses on the substance and the pan, and weights are added on the other pan to maintain the equilibrium. When the pan ceases to increase in weight the readings are noted and the temperature of the steam read on a thermometer. During the final weighing the steam is all to enter die chamber through a narrow escape-tube so as not to disturb the pan. The weight becomes practically constant in four or five minutes though a very slow increase of about 1 milligrams per hour may be observed due to radiation. The difference between the two weighings gives the weight of steam condensed. If W is the weight, of the substance, w the increase in weight of the pan, $ t the initial temperature of the enclosure, $ s the tempera- ture of steam, k the thermal capacity of the pan, L the latent heat of steam, s the required specific heat, then k is determined from a preliminary experiment without any substance on the pan and thus the specific heat of the substance is found. For great accuracy •various precautions and corrections are neces- sary. Steam condenses on the suspending wire where it leaves the chamber and then surface tension renders accurate weighing difficult. A thin spiral of platinum wire in which a current Hows, usually surrounds the suspending wire just above the opening, and is made to glow so that the heat developed is just sufficient to prevent con- densation, A rapid introduction of steam is necessary in the early s, for .steam also condenses on the pan due to radiation to the cold air and the chamber, thus causing error. This is, of course, partially balanced bv radiation from the steam to the substance later. 3G CAL0RIMETRY [< a ip. Further w does noi accurate!}' represent the weight of si earn con- densed since the first weight is taken in air at 0-^Q and the second in steam at Ss°C. All the weighings must be reduced to vacuum and then the increase in weight calculated. Specific heat of raw substances ran be found by this method since small quantities ui the substance are needed but a sensitive balance is indispen; The specific heat of liquids and powders can be found by enclosing them in glass or metal spheres whose thermal capacity is taken into account. Cases can also be similarly enclosed, but then the modified form of the apparatus — the differential calorimeter — is us< 12* The differential Steam Calorimeter, — -In this form invented by Prof. Joly in 1889, both the balance pans are made exactly similar and of equal thermal capacity and hang in the same steam-chamber (Fig. 5). The substance to be tested is placed on one pan and the ex- cess of steam condensing on this pan over that on the other pan is entirely due to the substance. Thermal can. i of the pans, radiation from them and all other sources of error common to them are eliminated, the substance bearing only its own share of the error. The chief use of this apparatus, hoy consists in the determination of the spe- cific heat of gases at constant volume. The pans are then replaced by two equal hollow spheres of copper furnished with "catch-waters" (shown in the figure), One sphere is filled with the dried S mental gas at any desired sure while the other is empty. These spheres are counterpoised by adding necessary weights m which represent the mass of the contained gas. St* is admitted and condenses on the pans. A larger amount of steam condenses on the sphere containing the gas, the excess, say w t giving" the amount of steam required by the gas. Now the specific heat, at constant volume c„ may be calculated from the equation 'fy— e t )=wLt (13) e $& 3 are the final and initial temperatures of the chamber. Prof. Joly used copper spheres of diameter 6.7 cm. and weighing pa. and employed gases at different pressures. Com tio applied for the following : — 1. The expansion of the sphere due to increased temperature the consequent, work done by t,lte gas in expanding to this volume. : - itiHl St. 'run Colorimeter. ll.j iiUiCTRl'CA I . METHODS o7 2. The expansion of the sphere due to the increased pressure mi the L;as at the higher temperature. le thermal effect of this stretching of the material of which the sphere is made. 4. The increased buoyancy of. the sphere due to its increased tie as die higher temperature. rhe unequal thermal capacities of the spheres. 6. The reduction of the weight of water condensed to its weight acu Dewar has devised calorimeters based on an analogous principle in lie employed a liquefied gas as the calorimetric substance. Tiie t. to be measured is applied to the liquefied gas whereby the liquid i vaporates absorbing its latent heat and the volume F of gas thus duced is measured. The heat communicated to the liquid is then en by V pL where p is the density of trie vapour and /. the latent at of vaporization of the substance. Using liquid oxygen and liquid d ogen the apparatus can be adopted for very low temperatures. 1 1. the case of hydrogen 1 c.e. of vapour at N.T.P. corresponds to a small quantity of heat (about 1/100 calorie). This method has id for measuring the specific heat down to very low tempe- The experimental substance (solid or liquid) is first kept in itant temperature bath (say 0°C) and then dropped into the containing liquid oxygen or liquid hydrogen. 4, ELECTRICAL METHODS 13. The electrical method was first employed by Joule in his npts to determine the mechanical equivalent of heat. The trical methods at present available may be subdivided into two : — (1) Method based on the observation of rise of tempera: :'') Method employing the steady-flow electric calorimeter. We shall first consider the application of these methods to liquids mse historically the method was first applied to them. 14. Methods Based on the Rise of Temperature. — Following Joule this method was adopted by many workers the chief among them being Griffiths, Schuster and Gannon, W. R. Bousfield and W. E, BouSfield. They employed Bliss method for determining tile mecha- lical equivalent of heat and found that it was capable of the highest ■ uracy. The same arrangements may be employed for finding the he heat of liquids. The principle of the method is to generate heat by passing a renl through a conducting wire. If i is the current through the wire of resistance R and E the potential difference across its ends, the spent in a time t seconds is Eit ergs, provided K and i are ed in electromagnetic units. If this raises the temperature ol M grams of a substance by M°, the specific heat s of the substance is given by the relation EU—JMsM, (14) ere / is the mechanical equivalent of heat (see Chap, J if). If E 38 CALORIMK'fltV [chap. is expressed in volts and i in amperes the energy spent is give | anles (1 Joule = 10 7 ergs.). Any two of the quantities E, i and R may be measured, Lhus giving three methods. Griffiths, in his determination of the sp neat of water, chose to measure E and R which is rather difficult for R must be measured during the heating experiment, GiiffiLhs' work is important since it first established the fact that the electrical method can accurately give the value of J in absolute units, Schuster and Gannon measured E and L Jaeger and Steinwehr have applied this method Lo determine the mechanical equivalent of heat and hence also the specific heat at different temperatures. They employed a large mass of water (50 kg.) and consequently the thermal capacity, of the vessel was only about 1% of that of the contained water. A section of their apparatus is shown in Fig. 6. AA is the cylindrical copper calorimeter lying on its side and properly insulated from the surrounding constant- temperature bath B. On the upper side at O there is a hole for the introduction of the heating coil H t the resistance thermometer and the shaft, t which drives the stirrer SS. A _nt. of about 10 amperes was allowed to flow for six' minutes through the const a n- tan heater H of S ohms resis tance and the rise in tempe- rature was about 1.4 °C In v experiments an accuracy in 10,000 was aimed at and hence the results are very reliable, IS. The Method of Steady-flow Electric Calori- meter,— Great accuracy was ained by Callendar and Barnes by using the steady- How electric calorimeter shown in Fig, 7. A steady current of the experimental liquid Sowing through the narrow glass-tube /, about 2 mm, in diameter, is heated by an electric current flowing through the central conductor of platinum. The steady difference of tempera- ture SB between the inflowing and outflowing water is measured by a pair of platinum thermometers Ft, Pt at each end connected differ- entially in the opposite arms of a bridge of Callendar and Griffiths' type, 'The bulb of each thermometer is surrounded by a thick copper tube of negligible resistance attached to the central conductor. This on account of its good conductivity keeps the whole bulb at the temperature of the adjacent water, and due to its Ion- resistance prevents the generation of any appreciable amount of heat by the Fig, 6. — A Section of jaeger and Steinwehr' 5 Calorimeter. STEADY-FiLaW ELECTRIC CALORIMETER OV current near the thermometer. The leads L, L and P., P are attached to this tube of copper, die former for introducing the heating current and the latter for measuring the potential difference across ihe central uctor in terms of a standard cell by means of an accurately Fig. 7.— Steady-flow Electric Calorimeter. calibrated potentiometer. The potentiometer also serves to measure the healing current i by measuring the potential difference across a standard resistance included in the same circuit. In order to diminish the external loss of heat the flow tube is enclosed in a herme- tically sealed glass vacuum jacket surrounded by a constant tempe- rature bath. Neglecting small corrections the general equation is Eit=JMs (ds-dj+jkt, .... (15) employing the same notation as before, where h denotes the heat loss per second on account of radiation, and & if &<, the temperatures of inflowing and outflowing water. The time of flow t in these experi- ments was about 20 minutes and was recorded automatically on an electric chronograph reading to 0.0 1 sec. The mass of water M was measured by collecting the outflowing water and was about 500 gm. The difference in temperature Q*—9t was from 8° to 10 D C and was accurately read to -001 n C. The heat loss h was very small and regular, and was determined and eliminated by suitably adjusting the i trie current so as to secure the same rise of temperature for different rates of flow of the liquid. Thus for two rates of flow we have E^t=JM v t{8 t -d x ) -f- Jht, •' J iM x -M t ) [Ot-W Since the temperatures at every point of the apparatus are the same in both experiments the heat loss h must also be the same. The i ecific heat s thus determined is the average specific heat for the interval W and may be taken as the specific heat at the middle of the interval. The great advantage of this steady-flow electric method is that no correction is necessary for die thermal capacity of the calorimeter 40 CALORIMri'RY [CHAP. since there is no change of temperature in an) part of the instrument Care must, however, be taken to secure perfeci steadiness, as it is practically impossible to correct for unsteady conditions. Further, since all condition?, are Steady, the observations cart he taken with tiie highest degree of accuracy. There is no question of thermo- metric lag. It is essential, however, that the current of water be thoroughly mixed otherwise temperature over a cross-section of the tube will' not be uniform. This is secured by having! the central Wictor in the form of a spiral instead of a straight wire, Callcndar and Barnes used this method to find the specific heat of water at various temperatures. Their results are discussed in the next, section- Callendar found the specific heat of mercury by this method. The central conducting wire was dispensed with, die flowing mercury itself serving as the conductor. Griffiths employed this method to determine the specific heat of aniline over the range i to 50°C. 16. Specific Heat of Water. — In ordinary calorimtftric experi- ments the specific heat of water is assumed constant, at all tempera- tures and equal to unity. Accurate investigations of the last section, show that it varies with temperature. The first accurate experiments in this connection were those of Rowland in connection with his deter- mination of the mi' ical equiv: leni " : heat (Chap. 111). He argued e specific heat of water at all temperatures were constant this mechanical equivalent must come out a constant quantity even' used water at different temperatures. The variation in the value irk heat. ?■ at different le\ wres. mp. Callendar Jaeger and Osborne Specific heat arnes Stemi Stimson & 5n int. Joules (O, S, & G.) 1.0093 (1.005) 1 .0076 1 4.2169 .1 1.00-f7 1.0029 1.0O39 1.2014 10 1.0019 1.0013 1.0015 4.1914 15 1 .0000 1.0000 oooo 11850 20 0.9988 0.9990 9991 4.1811 25 0.9980 0.9963 0.9fl 4.1788 ' 30 0.997ti 0.9979 9982 4.1777 35 0.9973 0.9978 0.9982 4.17. 40 0.9973 0.99R1 0.9983 4.1778 45 0.9975 0.9987 0.9985 4.1787 50 0.9978 0,9996 0.9988 4.1799 60 0.9987 0.9997 4.1836 70 1.0000 1.0009 4.1888 80 LOO 17 1.0025 4.1956 90 1.0036 1.0046 4.2043 100 L0057 1.0072 4.2152 iu NERNST VACUUM CALORIMETER 41 The oilier accurate experiments on the subject are those sndar and Barnes (sec. 15) and of Jaeger and . Steinwehr (sec- 14). Both of them determined accurately the specific heat of water at us temperatures, Their values are given in Table 1 together with the value;, obtained recently by Osborne, Stimson and Ginnings at; the National Bureau of Standards, Washington. In column 5 the specific heat is expressed in international Joules* per gram per °C. 40 s ST TSUPERAI b*RE K II-!..' Ft- 8.— Specific Heat Curve far Water. he results of all these three investigations arc plotted in Fig. 8. It will .he seen thai the values obtained by Callendar and Barnes lie somewhat wide of the others and appear to be less reliable chiefly on account of the uncertainty in the values of the electrical units employed. From these curves it is evident that water has a minimum ific heat at about 34 n C. It is on account of this variation that on IB the calorie was defined with respect to 15 C C. Specific Heat of Solids 17. Rise of Temperature Method.— The electrical method was first applied to solids by Gaede in 1902. E. H. Griffiths and E. Griffiths determined the specific heat of many metals over the range - 160° to | 100°C. The substance was used in tire form of a calori- meter and was first cooled below the desired temperature. Electrical energy was utilised in heating the calorimeter and the temperature led by a resistance: thermometer. The calorimeter was enclosed in a constant temperature bath whose temperature was kept constant to 1 /100th of a degree. Correction was applied for the heat lost by radiation, »yl8. Nernst Vaeuam Calorimeter.— A different form of the appara- tus, known as the vacuum calorimeter, was used by Nernst and Lindemann for measuring die specific heat at very, low temperatures. This differed from Gaede's form essentially in having the calorimeter suspended in vacuum. The results achieved with its aid are of great theoretical importance and hence their apparatus will be considered * 1 /Int. Jodie = 1.00041 X 10 7 ergs. CALORJMETRY CHA '. in some detail, T?or good conducting solids such as metals the calorimeter shown in Fig, 9(a) was used. The substance whose specific heat is to be determined is shaped into a cylinder G, having a cylindrical hole drilled almost through its entire length, and a closely fitting plug P made for it from the same material. The substance here acts as its own calorimeter. The plug is wound over with a p c fa) ( b) (C5 Fig. 9.— Nernst Vacuum Calorimeter, spiral wire of purest platinum (shown dotted in the figure) which is insulated from it by means of thin paraffined paper, and finally liquid paraffin is poured over it. The upper part of the plug is somewhat thicker than the lower part, thus a good thermal contact is obtained. The calorimeter K thus constructed Is suspended inside a pear-shaped glass bulb [shown at (b)] which can be filled with any gas or evacuated. The whole can be surrounded by suitable low temperature batlis such as liquid air or liquid hydrogen. The platinum spiral, which serves both as electric heater and resistance thermometer, is connected in scries with the battery B, resistance r and a precision ammeter A, the voltmeter V indicating the potential difference across the spiral. In order to bring the calorimeter to the desired tempeiature of experiment., hydrogen which is a good conductor of heat was first admitted into the pear-shaped vessel and the latter surrounded In a suitable bath. Next the vessel was completely evacuated so that the heat losses from conduction and radiation were almost entirely eli- minated. Tn addition it was surrounded by liquid air or liquid hydrogen. it. RESULTS or EARLY EXPERIMENTS experiment a current was allowed to flow through xnds and the voltage across it was adjusted to be denote To carry out an the heater for I seconds and the voltage constant by varying the resistance r. If Rf, Ri and i fs the final and initial values of the resistance of the heater and the enl through it respectively and E the constant potential difference. Thus an observation of it, i t and E gives R f and R, » and from a previous determination of the resistance oE the platinum spiral at various temperatures the rise in lemperature 80 can be found. The energy supplied electrically is Eit where i is the average value of the current. Now if M is the mass of the substance forming the calori- meter, s its specific heal, we have, Eit = JMsM -f h. . . . . (15) This gives the specific beat at a single temperature since BO is usually 1° or 2°. The heat capacity of the paper and paraffin can be found and eliminated by taking different amounts of the substance and at the same time arranging that the temperature rise is the same The heat loss h is very small and is determined and accounted for I - observing the rare of cooling before and alter the experiment. For non-conducting solids' the calorimeter shown in Fig. 9 (c) was employed. The heating w T ire was wound over a cylindrical silver vessel D and the whole covered with silver foil to diminish heat loss. This foil was soldered at the bottom of the cylinder as indicated. The solid whose specific heat is required was placed inside the silver cylinder and the latter closed with the lid. The silver on account of its high conductivity keeps the temperature, uniform and this is further secured by filling the cylinder with air through the tube in the lid. The tube is then closed with a drop of solder so thai it mav be gas-tight. It is absolutely necessary that air should be sent inside the vessel to facilitate equalisation of tern pern lure throughout, the experimental substance. Liquids and gases can be similarly admitted into the cylinder and their specific heat determined. 19. Results of Early Experiments.— In 1819 Dulong* and Petit from their investigations concluded that the product of atomic, weight and specific heat 1 xf/as constant for many substances, or in other words, ims of all substances have the same capacity for heat. Regnault from his own researches found that for ordinary substances the mean value of the constant was 6.38 with extremes of 6.76 and 5.7. A more accurate value of the constant can be obtained from the kinetic theory (Chap. III). The atomic heat at constant volume is shjwn there' to be equal to 31? = 5.955. According to Richarz, the value ::T the ratio Cp/c e for many substances lies between 1.01 and 1.04, hence the atomic heat at constant pressure, the quantity commonly determined should lie between 6,01 and 6.19. This law T is of great * Pierre Louis Dulongr (1785-1838), a distinguished French scientist who lost an eye and a finger owing to the explosion of some nitrogen chloride which he discovered. CAI.ORIMETRV CHAP. aiming atomic weights. Tn illustration of the law table 4 - is adde . Ta ble 2 — Illustration of Dnlong and Peti Element am M; '_ AJuminiitiii Iron led Copper Zinc Silver lium Antimony LI Atomic Weight Mean specific . (1) IlC Hi heal : (2) 23*00 0.307 7.06 L32 0.247 li.nii 27.1 0.2175 5,83 55 0.110 6.14 .68 0,1092 6.41 63.57 0.0930 6&,37 0.0939 107.0 0.0559 6.03 11. 0.05 6.26 118.7 0.0556 121.8 0.05 6,10 [95 0.0318 6.21 197.2 0.031 8.10 0.0310 209 0.0299 ies= 6.24 ,n mn enunciate aw concerning molecular heat and onstant The - ; ll "' : " ' mounds to another In illustration of the tbkf 3 is added. —Moteo at of Oxides. Compound i tfic heat Cr s O a Bi a O, Moleculai . Secular -dit heat (2) 0.1700 0.1796 0.1277 0.0901 0.0605 159.8 192.0 197,8 287.fi 27.2 27. J . 25.9 Mean value = 26.8 • ! :en f rom Vol S n 193 ' lake!1 : Experimental ! ' g p , 200, ■". TWO St'I CI! [CI Limann's law can be considered as a particular case oE die following law : 1 i ular heat of a compound may be considered is the the atomic heats ol Its constituents. Thus if a com- pound has the composition A a "&i,G c T)j its molecular heat C t is given n the relation C p =aC,, , D . . . . (17) B, C, D stand for the different types of atoms composing the compound and C pjLt C p1i , etc., their atomic heats given by mlong and Petit's law. The law is ol much use in evaluating the nolecular he. [lain substances, 20. Variation of Specific Heat with Temperature. — The Specific heal determined by the foregoing methods is not found to be a constant juantity. For solids and liquids the effect of pressure on specific i rather small. The effect of temperature is however very d. Increase of temperature invariably -increases the specific feat, while the decrease of temperature lowers it. In fact the atomic lea solid almost vanishes at the absolute zero and gradually increases with rise of temperature reaching asymptotically the Dulong bid Petit's value of 3JS at a sufficiently "high temperature which is Iiflerent for different substances. This variation for silver is illus- d in Table 4. —A torn ic heat of silver at different temperatures. Temp, in c Iv 1.35 S 10 20 36.16 0. i:'V 0.00509 0.0475 0.399 1.694 Temp, in °K. Atomic heat 55.88 74.56 103.1 I 205,3 3.186 :.u;'-| 5.373 5.605 The asymptotic value oF $R can be accounted 1 for by the Kinetic Theory (Chap. Ill) . The variation with temperature has been successfully explained by the quantum theory of specific heats and in particular by the Debye's theory of specific heat which is how beyond the scope of this book. 21. Two Specific Heats of a Gas. — The specific heat of a gas. as of solids and liquids,, may be defined as the ratio of die heat bed to the rise in temperature, taking a unil mass of the gas. A little consideration will show thai Emition requires to be 1 'ne a quantity of gas to be suddenly compressed. The ' ii : oJ the gas will be found to rise, though no heat has been added. The rati •' added/increase in tempera' v. ic heat, vanishes. Again let this com] air expand ; ]]'.. ; :\ roolrng v. ;ikc place. Tins is just prevented bv applying some heat to the gas. In this en.se the ratio, heat ad 415 CALORIMETRY CHAP. EXPERIMENTS OF GAY-LUSSAC & JOULE 47 change in temperature becomes infinite. Thus we see that the original definition gives an infinite range o£ values for the specific m Hence external conditions are of paramount importance in deter- mining- the specific heat of gases. It has become customary to ak of two specific heats o£ a gas : the specific heat at constant volume denoted by C, and the specific heai at constant pressure denoted by tip. In the former process the gas is maintained at constant volume so that the whole heat applied g es to increase the internal energy of the gas. In the latter case die gas is allowed to expand against a constant pressure and in so doing it does external work. This work is obtained by using up part of the heat energy applied ^ to the gas. Hence the specific heat at constant pressure is necessarily greater than the specific heat at constant volume by an amount which is simply equal to the thermal equivalent of' the/ work done by the gas. Let us assume that the gas is perfect, i.e., its molecules exert no influence on one another. This is approximately true for the permanent gases as Joule's experiment (Sec. 22) shows. Hence in this expansion no internal work -against molecular attractions is done by the gas and the excess of heat supplied in the second case is simply the thermal equivalent of the external work. Consider the gas enclosed in a vessel of any shape and suppose the walls of the vessel can expand wards. Lei &A denote an element of area of the walls and Bx the nice traversed hy, it measured along its outward drawn normal. the work done by the gas in this expansion against a cons- ' pressure :.j^; and for die expansion of the enth the work is equal to % p&AM = pl&AJIx — p&V where lie volume of the gas. Suppose a gramme- ping a volume V t at temperature T°K and pressure to a volume V-, at temperature (T -|- 1) Cl K the og constant Let Mc p = C p , Mc p = C, where M h the molecular weight of die gas. C* and C, may be called trram- 22. Gay-Lussac first car- internal ■nol.- molar specific heats, expanding from V x to l-% is The work done by the gas in P(V S - V t ) ^ R [ (T +]}-T]=R J . . (18) y the gas laws. The thermal equivalent* of this work is T(C* — C ) ■ . . (19) and C. measured in Hence JfC^-C,) =Rf, Or, if either R is expressed in calories, or C* . : ■•■■ we get C P -C S =R. This relation was first deduced by R. Mayer. if] oS^mS^. FirHt Taw ° f T^^anfe which will be diseased tThia is true for a perfect sas only. For real gases it cau be shown that Cp ~ CB = T \dT) L ,\ arJpWhere the differentials I be evaluated from the- Experiments of Gay-Lussac;!: and Joule. nc«| out experiments to determine whether a gas does any internal in expanding. He allowed gas contained in a vessel at a high o expand into an evacuated vessel and observed the fall of [en perature in one and the rise of temperature in the other vessel. is called free expansion or Joule expansion. In this the gas as whole does no external work while its volume increases, and the work done will be that against molecular attractions which is nal work. Somewhat later Joule employed a similar iratus but he immersed the vessel in calorimeters. His apparatus actual equation o£ state (Chapter IV) for the gas. Joule's experiment with Joulc:^ experiment with one calorimeter, two calorimeters. Fig. 10 Fig. 11 indicated in Figs. 10 and 11. The two vessels A and B corn- " i a ted with each other through a tube furnished with a stop-cock A was filled with dry air at 22 atmospheres while B was exhausted. prat the whole apparatus was placed in a single calorimeter (Fig. 10) md the stop-cock C opened. No change in temperature of die water fas observed showing that no internal work against molecular attrnc- done I the gas in expanding. To investigate the point iriher, the parts A, B/C were placed in separate vessels (Fig. II;, attaining water whose temperature could he read by sensitive lermometers. On opening the stop-cock C the air expanded into B id the temperature of the vessel surrounding A fell while the tern- >eratures of B and C rose. It was found that the heat lost by A /as exactly equal to the sum of the heats gained by B and C, jus the total change in the internal energy of the gas during expansion zero. *.e\,\l— J — where U is the internal energy and P ic volume. This is called Joule's law or Mayer's hypothesis. This hows clearly that no internal work is done by a gas in expanding. [oule's law holds only for the perfect gas to winch, the permanent jases of Nature like helium, hydrogen, etc. approximate. For further J Louis Joseph Gay-Lussac (1778-1850) was a distinguished French scienti fho investigated on the expansion of gases. He was interested in aviation and 1804 made a balloon ascent for the purpose of making experiments,. He was a peer of France. CALORiM ' [CHAP, iSUKEMENT OP C* 49 discussion see Chap, \ I, sec. lf> and Chap. X.. sec 3. As a n of fact, a slight fall in temperature should be observable in Ji experiment with one calorimeter but on account of the I. capacity of the calorimeter it escaped detection, 23. Adiabatic Transformations. — When the pressure and volume of a substance change but no heat is allowed to enter or leave it, the transformations are said to be adiabatic (a = not, dia = thro bates = heat, i.e» 3 heat not passing through) • In an isd hermal i h thi temperature is kept constam by adding heat to or taking it away from the substance. Consider an amount o[ heat 8Q applied to a perfect gas. This is spent in raising the temperature of the gas and in doing external work. H we consider i jramnn olecule of the gas, the former is equal to C r d'f and the latter equal ro pdV/J, both in calories. Hei,. MZ = C,dT + pdV/J, or, if Sf) and C., are measured in ergs, = C,dT + pdF. ... This equation combined with pV = RTyn\l give the solution oi all problems on perfect gases,* In an adiabatic transformation 80 = 0. Therei C B dT- r pdV = i in order to find a relation between p and V we must eliminate M (21) I jas equation pV = RT. DifTerentii Vdp = RdT i dT from (22) in (21) , we get and re pi ii by C p -C„. C,Vdp + C p pdV = 0, •'ting C p jC 9 by y we obtain dV i>« which on integration yields log p + y log V = constant, or — constant, is the adiabatic relation between p and V for a perfeel gas. find the adiabatic relation between 7" and V or bet reen /'and p we must respectively eliminate p or P betwei n ind the gas equal j-pv-i__ constant (24) ■ — constat 1 1 i In ■"." bRT, and CV-C I substitttl equation oi state mid the true value oi C,-C„ We ion between />, F and 7 for real i^aes. , ._Drv air enclosed at 25 e C and ai atmospheric pressure ,, 1 ,.] to half its volume. Find (a) the resulting the resulting pressure. Assume y = 1.40. !4) T 1 = T(7/7 1 )-- 1 -(273+25) (2)^ ~393«K. : 'I \V x )v--pV^ ut =2-64 atm. 24. Experimental Methods*.— Let us now consider the expej ods oi Ending the specific heat of gases. Since for ises C„ -C B — R, a knowledge of one of the specific heats other. Again, if we dejpmine y, i.e., C I ie above and C B . Hence the experimental methods ivided into three classes : (a) The measurement oh (.',, (b) The measurement oE C,, (c) The determination of y. : MEASUREMENT OF C p 25. The specific heat at constant pressure has been found either by the Method o£ Mixtures or by the Constant-Flo thod. The principles of these methods have already been explained. 26. Method of Mixtures— Regnault's Apparatus. The method was first applied to gases by Lavoisier and Laplace. Improvements were GE2T I •!;■ imauh's Apparatus for CV. '■ The reader will find a very good account of these meQwxLg in Partingti n Heai oj Ga 50 CAUHUMi CHAP. made later by Delatoche and Berard, Haycroft and Regnautt. Regnault with his groat, experimental skill obtained results of high acy. His apparatus is indicated in Fiu;. 12. Pure dry gas was compressed in Lhe reservoir A which was immersed in a thermostat. Mi. reservoir was provided with a manometer (not shown). Gas could be allowed to flow through the stop-cock V at a uniform rate. This was effected by continuously adji Ele stop-cock V (shown ■ rely) so that the pressure indicated jry the manomel r R was constant, The gas then flowed through a spiral S [n : in a hoi nil-bath and then into the calorimeter C fir:: ; into lhe air. The gas acquired the temperature T oj rath and raised the temperature of the calorimeter, say, from (^ to i),. Tf m is the mass of the gas that flows into the calorimeter, r^ its specific heat, is given by (26) mo (r A t*)-»(V- w is the thermal capacity of the calorimeter and it* contents. I 1 1 .: : mass m of the gas was determined by Regnault as follows; — f€>r any pressure p he assumed that the weig lined in the reservoir at temperature was given bv the relation W(l + a&) =Ap+.r: C B t C were determined, from leliminary experi- rresponding to the observed dent is given \p> -f- Bp of the gas iras found.* i>i gain <■'• and radiation which have to be found out .tare of the calorimeter before and the experiment, and taking the avea 27. Other Experiments, The experiments of Wiedemann v similar to those of Regnault. Lussana devised a high- ire apparatus in which the same amount of gas enclosed at a high pressure can he repeatedly healed and passed through the The principle employed is the same as in Regnault's iment. This apparatus can be used to find the specific heal o£ ery high pressure and possesses the advantage that the gas is not wasted. For determining lhe specific heat at high temper;) he experi- ment of Holborn and Henningmay he mentioned. Employing -.u liable floe trie heaiers, resistance thermometers and well-designed calori- * If the i . ; relation po = RT/M (p. 5), where v i* the specific volume inn: M its molecular weij assumed to hold, a simpler expression For W ijned. Let V denote the volume of the reservoir 1 a'szl/v, the density of the gas. Then W— V P = 3g£, and IY-W'=™ (p-f), where p is cxprtHsed in dynes and R in ergs. 'RT MEASUREMENT OF C, 51 able to find the specific heats of nitrogen, carbon i team up to 1400°C. The calorimeter corrections are uncertain. The same method has been used by Nernst the specific heat of ammonia up to 600 °C. 2ft. Constant-flow Method. For finding tire variation of specific ii i nperature the constant-flow method is most suitable, and used by Swan. The most recent form of the apparatus is thai used I !•. ill I'.' el and He use in finding the specific heat down to ow temperature and is in principle similar to that shown in 1 to rider test., previously brought to a steady temperature ugh a suitable bath, flows through the calorimeter in i. Inside the calorimeter it is heau-'d electrically by in' roil of constantan and thus the energy supplied can be • I i M iv, of the incoming and outgoing gases ranee thermometers. The specific heat, can be v an equation similar to (15). li at low 1 mperatures the gas was initially passed through i| i rature bath in which the calorimeter was also immersed. 'hum and other rare Scheel and lleuse modified o as to employ a closed circuit These experimenters 1 icasurements on various gases in the range 0° to - 180 o C. Foi measurement of specific heat at high pressure the constant" od has been employed by Holborn and Jakob and gives ih able results. (b) MI'J HODS BASED ON THE MEASUREMENT OF C v .Steam Calorimeter. The direct determination of C. is best ol Joly's steam calorimeter (sec. 12). The method th< periment and the necessary details will be found on. 1J - 1 ' i ■ • c |-l -i, Mm \ 32 CAI-ORIMKTRY [CHAP. 30. Explosion Method* Following the work of Bunsen, Vieille and others, Pier improved the explosion method and devised the modern explosion bomb indicated in Fig. 13. Immersed in the water- bath B is a steel-bomb A which has a side tube M through which the. bomb can be evacuated and various gases introduced at the desired partial pressures. S is a corrugated steel membrane closing an open- ing in the bomb and carrying a mirror S. Light reflected from the mirror (alls on the photographic film F which revolves on a drum 3y applying various known static pressures and noting the deflection of the light spot, the pressure attained in any experiment can he found simply from the record on the film, Any explosion mixture, say, a mixture of hydrogen and oxygen together with tlit; inert ^as whose specific heat is required, is introduced in the bomb, L>y 'inert' is meant any gas which will not take part in the reaction either from want of chemical affinity or due to its presence in excess. The partial pressures of the various gases are known. The explosion is started by means of electric sparks and the final pressure reached is observed. This takes about .01 sec. The calculation may be easily made. If the vessel were allowed to cool to the initial temperature T 2 (absolute) , suppose it. would record the pressure pn. The maximum temperature T-, reached during the explosion is calculated from the value of p x , for, from^the gas , since the volume remains constant, r[ T — Pit Or ii ite the initial pressure and e. the ratio of the final to tb 1 number of molecules (owing to the explosion the total m i of li = e Pi [ '" volume and temperature are me] ■r —P* t 7" (28) where P is the ratio of explosion pressure to initial pressure. The relation connecting the specific heats and the heat of reaction is mih- (Tx-Tz) [mC ar -\-nC H ] f . . . (29) where m is the number of gramme-molecules of the reaction products ; n the number of gramme-molecules of the inert gas; C vr} C re ,resper- their mean molar specific heats over the range (T* - 7Y) , and Q 2 the heat of reaction for the explosion mixture at T a °. C> £ is generally known from thermo-chemical data. We can determine C pr by explod- ing either with different amounts of a gas whose variation of specific heat with temperature is known, or with different quantities of argon, a substance whose specific heat is constant. Then the same reacting gases may be exploded with any inert gas, and knowing C v r , we can find C Di for the inert gas. The method is very suitable for measurements of specific heat at high temperatures and has been used to about 3000°C but suffers from the disadvantage that directly it gives only the average values H I DETERMINATION OF y 58 wide range and not the specific heat at any temperature. iluv, corrections are necessary for loss of heat, effects of dissoeia- n, incomplete combustion, etc. For argon. Pier found that the heat does not vary with temperature. 3L list's vacuum-calorimeter method is employed to find ii low temperatures. The gas is enclosed in the calorimeter j. Eucken in this way found the specific heat of hydrogen . :>°K and obtained interesting results. METHODS BASED ON DETERMINATION OF y 32. As already pointed out this is an indirect method of finding the s iecil c heats of gases. Though indirect it is capable of the highest v so that the modern accepted values of specific heats are on the values of y thus obtained. The methods for measuring y, the ratio of die two specific heats, ii !'• ie classified under two heads : (I) those depending on the ■<ic expansion or compression of a gas, (2) those depending on the ty of sound in the gas. We shall first consider the former, (1) Adiabatic Expansion Method 33- Experiments of Clement and Desoraws*— Clement and Desormes >: the first to find y by the adiab; tic expansion method. Their .inal apparatus has been considerably improved and is indicated in Fig. II, A large flask A of about 28 litres capacity is closed I • a stop-cock M about 1.4 cm. in diameter. Tlie flask is iiumected to the manometer P t -P,.[ by means of a side-tube and is with cotton wool to avoid loss of heat. First the flask is tialJy evacuated and the pressure p t recorded by the manometer, is observed. The stop-cock M is then opened and quickly closed. Air ics into the flask till i he -pressure inside and ide becomes equal, i 1 i ■: ■ process is adiabatic die loss of heat in the short interval for which the stop-cock M is • ii may be neglee The temperature of the air in the flask rises on unt of the inrush of* external air and the are becomes atmos- pheric. The flask is next allowed to cool to the imperature of the sur- • 'i idings when the water in I he :.■ On ter rises and finally indicates the ' Fig. 14.- -Clement and Desormes' appai - 1 ™ 54 GAL.OKIMETRY | GfiAPi Let the atmospheric pressure be p A and the specific volumes of air at the pressures, p- t p Ai p f , be respectively v it v A , i>/. The first: process is adiabatic and hence we have, assuming the gas lo be perfect, p z V-'---p.4l<A v TO Since the final temperature is the same as the initial, we have, considering 1 gram of the gas, SpFi^PfVf : Again, v A --v ; . (32) amse the volume of tile manometer tube is negligible compared with that of the flask and hence there will s no appreciable .. auge in the specific volume of the gas due of liquid in the manometer. Combining (30), {81), and (32), we have. Pa \pfh yJSiCJEiiL. log pi iogp f II, as usual j the changes in pressure arc small, ' P-P, f From th irements of Clement and Disomies, Laplace deduced lu< of y to be IM In this experiment there is a source We have 1 that the pressure inside has become atmospheric when the stopcock M is closed- Actually, however, osdllatio on 'Lint of the kinetic energy more air first rushes in than would make pressure just atmospheric, and hence the pressure, inside becomes greater than p A . Next some air rushes out till the pressure inside « is less than p A and so on. After several such overshooting? the pressure p A h attained. This takes considerable time and, as a matter of fact, this to-and-fro motion has not subsided when t,he stop-cock is closed. It must be closed at the instant when during an oscillation the pressure just becomes atmospheric. This is very difficult to secure and nence later investigators tried to avoid it by measuring the change in temperature resulting from adiabatic expansion. The scop-coi has not to be closed in this case. We shall consider shortly the experi- ments of Lummer and Pringsheim and of Partington and Shilling baged on tills principle. We have assumed above that the incoming air has the same temperature as the air in the flask initially. To avoid correction in v it is not so, it Is better to start with compressed gas in the flask. when a U-tube manometer must be used in place of t^P^. Further care must be taken to use perfectly dry air lor y is appreciably different for moist air. Consequently sulphuric acid is generally used as the liquid in the manometer. 54. Experiments of Joule, Lummer and Fringsheun, and Partington.' — Joule was the first to study the change in temperature by EXPERIMENTS OF "PARTINGTON Hllnbatic expansion or compression. Various investigators later i mi>!o\cd this method to determine y . Air was compressed m a vessel ,1 it s temperature and pressure observed. It was then allowed to i In pressure and temperature before expansion, p 2i quantities after the expansion, we have, from (25) .[ suddenly to atmospheric pressure and the change in tempera- noted. Trie calculations can be easily made. If p tl 7\ denote T» the same l-v n (l-y) log ! - " 7", 8^8 Pa £ 1 tog Pi- -1°$ Pi " Y (log Pi - tog Pi) - C lo S T i - J °g r a) y can be calculated, Lummer and Pringsheim made considerable improvements m the ratus for determining y by this method. They employed a 90-litre here and measured the change in temperature by the change topper sph... n i. sistance of a thin bolometer wire hanging atthe centre oJ this A Thomson galvanometer having a period ot 4 sec was Certain errors arc, however, inherent, in . . null instrument. 1 1 ir apparatus ill llllll .1 LLia. . In order to eliminate these errors Partington has further improved pparatus. He used a large expansion vessel (130 litres capacity) : bolometer of very thin platinum wire (.001 to .002 mm. m dta- SfFI T PJ 5 ft \ j: a • \ii iftvwvvv— iv ■ .■•.-.- • - i / Fig. 15.— Partington's Appafettis i witli compensating leads ; thus there was no lag. Further an ♦This is true for a perfect gas only. For real gases it requires mmiifi cation. V 56 CALORUI' CHAP., Einthoven string galvunor pable of recording temperature in .01 sec. is used so that a detailed record of changes in tempera t Lire of the gas during and after expansion is obtained. His apparatus is indicaLed in Fig, 15, The vessel A is provided with the expansion valve- C which can be manipulated by means of the spring P and whose size can also be varied, A is connected to the sulphuric acid vi mometer M, the mercury manometer m, and the drying tubes F. Thus carefully purified air enters A. Further the vessel A is kpt immersed in a water-bath which is kept stirred by S. B is the bolo- meter wire (shown separately, in the figure^ and is connected In one arm of a Wheatstone bridge. G is the string galvanometer. The initial temperature was read on a car standardised mercury thermometer T immersed in the bath and was given correct to .01°. Then the resistance in one arm of the Wheatstone bridge lowered to give some deflection in the galvanometer. It was so arranged by trials that immediately alter expansion this deflection was reduced to zero. After the expansion experiment some ice was continuously added to the bath to keep its temperature constant and equal to than immediately after expansion. This was ascertained by keeping the galvanometer deflection steadily at zero, and the tem- perature of the bath was again read on the same mercury thermo- meter. .11' the aperture is too large, oscillations of the gas take place and •ilvanometer deflection is not quite steady, the initial deflection being somewh tcr than the true value. If the aperture is .too narrow, prolonged expansion resull be process is not adiabatic. Tn pr; rture was gradually diminished and when over- shooting was eliminated the deflection was instantaneous and per steady. The atmospheric pressure was read on a Fortin's barometer and y calculated from the foregoing formula, y was found to be 1.4034* at 17°C. This method cannot be used at high temperatures since it is impossible to determine accurately the cooling correction. 35. Ruchardt's Experiment. — A simple method for determining y, which is suitable for class-room demonstration, has been described by E. Riidiarrit, The apparatus consists of a Large glass bottle V (Fig. 16) fitted air-tight with a glass-tube at the top and a stop-cock H at the bottom. The glass-tube lias a very uniform bore in which a steel ball of mass m fits very accurately- If the ball is dropped into the lube, it begins to oscillate up and down and comes to rest after a few oscillations. If the period of oscillation be determined with a stop-watch, y can be easily calculated Let A be the cross-section of the glass tube, v die volume of the bottle, b the barometric pressure, and p the pressure in the flask. Then in the equilibrium position ,-I + 3 * Using this, the velocity of sound in dry air at 6"C was calculated fn a equation (3fi) to be 331.38 metres which is in close agreement with Hebb's mean value 331.41 m./sec. (Sec 37). 1 VELOCITY OF SOUND METHOD 57 ball now moves a distance x downwards it compresses the diabatically increasing the pressure to p + dp, hence ,tt equatum . diabatically ol motion is m - — Force of restitution = -Ad;>. pv* — constant, we have dv Ax ■ypy-vPT- (35) i =-Ax. d*x s •ii .• lich the period of oscillation comes T = 2 y = v, 4ir •■• i (W) Fig. -16-— Rfidiardfs Apparatus. pA*T* m I litis, knowing T_, p and' the constants of the apparatus, y can be aked. (2) Velocity of Sound Method 36 This method also depends upon the adiabatic expansion i • -,.'. , a gas but differs from the foregoing method in that direct measurement of changes in temperature or pressure need io be observed. The method has given us the most accurate data regarding specific heats Cor both high and low temperatures and so- all consider it in some detail. The velocity of sound in any Kind is given by the equation U — ^nrfe where E s p is the adiabatic elasticity of the fluid and density. For adiabatic changes in perfect gasesf pv v -. constant (p. 48) , hence Ef = - B ||fi} -tf tan (35). U V hp (38) Thus, if we determine the velocity of sound in the gas we car* 37. We may adopt either of the jpo following methods. The ;ite velocity of sound in the gas may be determined, or we may *See Barton, Sowd, , . . , ,_, ir real gases we must take into account the true equation of state [Lhap. c the value of \/.:pi'iiv), from that equation. In all accurate work d in. &s CALOHIMJE.TRY compare the velocity with that in another gas (say, air) which has been determined accurately by other methods. For our purpose we discard the large-scale determinations of the city of sound in air on account of the various defects inherent in them. The most, accurate direct determination of the velocity of ; 'itcl in air- was made by Hebb in 1905 by a method depending on reflection of sound of known frequency from parabolic mirrors, His mean value after employing ail corrections gives 331-41 r sec, as the. velocity of sound in air at 0°G and 760 mm. pressure. Now we must remember that practically all determinations of the velocity of sound in gases have been made in tubes, but the velocity in ;:. tube is not the same as in free space. Corrections have to be applied to reduce this velocity to that in open space as explained in the next section. In equation (37) the velocity in open space must substituted. Dixon has directly determined the velocity of sound in different gases from 15°C to 1000°C in a very satisfactory manner. 5 result may be employed to give y. Method based on the Measurement of Wavelength. 3S. Kundt's Tube. — Kundt first devised an apparatus by means of which he could find the velocity of sound in a gas. This consists simply of a glass tube about 1 metre in length and diari eter. One end of the tube was fitted with a mo opper, while through b SrE s dl^l U^ 17,—. Kundt's Double-tube A] loosely-fitting disc: carried by a glass or metal : ing-rod 11 clamped at its centre. Later Xundr employed the double-tube apparatus indicated in Fig. 17. Two tubes art; connected by means of the sounding-rod S which as ; l] .- 1 at distances one-quarter of its length from either end. The .bber corks., d, d in the tubes A and B provide the damping ngement. The pistons P. P can be moved to and fro lo bring die tubes in resonance with the rod S. Throughout, the length oi each tube is spread some light dust such as Ivcb podium powder or silica dust. One tube is filled with air and the other with the experi- mental gas. The sounding-rod S is excited by rubbing it at the re when the dust is thrown into violent agitation" at the anti- nodes and collects at the nodes. The distance between successive nodes equals ha If- wavelength, and knowing the frequency of the sound the velocity is easily obtained. The double form of the apparatus *l very convenient for comparing die velocity of sound in anv ihts ,'ith that in air for ' w . (39) VELOCITY OF SOUND METHOD 59 precautions are, however, necessary. The tube and the , us be perfectly dry. Carefully purified air must be used n for the various impurities must be made. Too much , not be used for excess of dust diminishes the ve loci y. in the diameter of the tube diminishes the velocity. m the velocity in the tube the velocity m open space must reed. Though mathematical equations giving the req m . on have been developed by Helmholtz, Kircbholt and others re not quite adequate and the best method is to express the U' in the tube as U' = U(l-kC), . . . ♦ (40) where V : velocity in open space, h -= a constant depending on the tube (its radius, thickness, the thermal conductivity, surface, frequency of the sound, etc.) C — a factor depending on the gas (its viscosity, density, ratio of specific heats, etc.) Kirchhoff showed that C = /fl'Ww)! here » = viscosity, p = density, Y r= ratio of specific heats and bte K being the thermal conductivity. For air from Hel ■■■ riments we know U } and by observing U[ and calculating C, k h tube is determined. This value of k is employed to give the veld-ay in open space for any other gas. Kundt and Warburg later employed this method to determine the velocity of sound in mercury vapour. One of the tubes contained ury vapour and was heated in an air-bath to about 800 C, The distance between two nodes was measured when the tube cooled. They found y— 1.666, Ramsay employed this method to find y for argon but on account of certain difficulties he got a low value. Behn and Geiger improved the apparatus considerably. They dispensed with the sounding rod and employed a sealed tube containing the experimental gas as the source of sound. This tube was clamped in its middle and was excited like the sounding-rod. The tube should be chosen properly and its length be adjustable so that the contained gas may give resonance with "the sound emitted by die rod. T he apparatus" was eminently suited for gases at high temperatures. This method was later>mployed by different investigators particularly by Partington and Shilling. * It can be easily shown from very simple considerations that for a mi* of perfect cases, _a_ = h + ii_ r— 1 yi— 1 72—1 •where f h £s are the partial pressures^ of these gases, P the total pressure and Y stands for the ratio C P /C r for the mixture. 60 CALOR1METRY [chap, 39. Experiments of Partington and Shilling. — These investigators determined the velocity of sound in various gases up to 10G0 D ~G bv a resonance method. The apparatus is diagram m a tically represented in Fig. 18. I'F is a silica tube 230 cm. long and wound over almost along its entire- length with heating coils. To this tube is attached it X a glass tube, MM, 150 cm. long. Inside the former is the piston 1 J of silica carried by the rod A, also of silica. BB is a steel tube joined to A by means of a cork. The tube BB carries a saddl 18.— Parting-tan and Shilling's ,'■• ing on a millimetre scale, thus die displacement of the piston Through this tube pass die thcimo couple leads to the potentiometer system £. The other end of the silica tube is closed by a telephone diaphragm T which can be moved by means of the screw Y. This end i losed gas-tight by means of the bell-jar J. . The diaphragm is excited by a valve oscillator V giving a note of Erequency 3000. D is a side-tube from which a rubber tube leads to the ear of the experimenter. X is ati asbestos phie to prevent radiation of heat to M. v The silica tube is filled with the experimental gas and maintained at Che desired temperature. The central tube AB is gradually moved • from 1} and the successive positions of the saddle I on the millimetre scale corresponding to a maximum sound in D are noted. The successive distances correspond to A/2 and knowing the frequency the velocity is determined. The position of T has Co be adjusted at different temperatures in order to give maximum sound in D when it will be at a distance A/2 from the "latter. nno ?°°k employed this method to find y for air and oxygen from SO to p3 £. His apparatus may be visualized if we imagine the hot-air bath of Kundt and Warburg to be r-.huvd Uv :i h; ...- flask containing liquid air. specific; heat of superheated vapour 01 ■ i 1 1 1 i 40. Specific heat of superheated or non-saturated vapour.— Reg- _.£ determined the specific heat of superheated or non-saturated id other vapours with an apparatus which wa itially liar to that shown in Fig. 12 (p. 49). Steam is superheated to by passing it through the spiral S in the oil-bath kept at a .Lure above I06°C, The superheated steam is next passed n constant pressure into the condenser kept immersed in the water trimeter, and the rise in the temperature of the latter from 8 X to a noted. If m h the mass of steam condensed, e p its mean specific heat . rant pressure between the temperature T and its condensing tnt 0° at constant pressure, and L the latent heat of steam at d°> the quantity of heat given by the steam is Ulcers) +mL+m{&-8J, This must equal a; (0s- fli) where w is the thermal capacity of the lori meter and its contents. Hence ]T-B)+mL+m{e-6 s )=w{6 s -e t ). . . (41) experiment is then repeated with another value of T and a ►n< ' n similar to the above obtained. Solving the two qu the wo unknowns c p and L are determined, Regnault thus found c t for steam between 225*C to 125 L 'C to be 0.48. In case of vapours of other liquids, the specific heat of the liquid must be taken into account in writing out the above equations. 41. Results.*— In the foregoing pages we have considered the various methods of finding specific heats. In tablef 5 (p. S2) we e the values which best represent Lire experimental data. We reserve our comments on these values for the next chapter. replaced by a long Dewar | Taken from TLmdhuch tier E.vpfn>r<e> k, Vol. 8, p. 33. 62 CAI-ORIMETRY [chap. Table 5. —Molar Heats in Calories at 20 °C and Atmospheric Pressure, Gas Cp e. Remarks Argon Helium 4.P7 2,98 1,666 > 4.97 2,98 1,666 t Monatomic tlrogen 6.865 4,88 1,408 1 Oxygen 7.03 5,035 1.396 Nitroj 5.93 4.955 1.402 | Nitric oxide 7.10 5,1.0 1.39 -Diatomic Hydrochloric acid , . 7.04 5,00 1.41 Carbon monoxide . . 6.97 4.98 1,40 Chlorine 8,29 6,15 1.35 Au 6.950 4.955 1.402 Carbon dioxide 8,83 6,80 1.299 I Sulphur dioxide . , Hydrogen sulphide 9.65 7,50 1.29 - Triatomjc 8.3 6.2 1.34 . 8,50 6,50 1.9] \ i' ili 12.355 10,30 1.20 lene ene 10.45 10.25 8,40 8,20 1,24 1.125 r Polyatomic 8.80 *&! 1.315 42. Special Calorimeters.— Various types of calorimeters have been devised for special purposes e.g., for the measurement of the heat of coin bastion, heat of chemical reaction, heat of dilution, etc., but they involve no new principles. Particular interest, however, attaches to the determination of heat of combustion in industries, for the value of fuel u judged mainly from its calorific value. This heat can be easily determined with the help of the calorimetric 'bomb'. Fig, 19 indicates Che calorimetric 'bomb*. It consists of a stout el-cylinder A fitted with a cover held down tightly bv suitable means. The cover has a milled-head screw valve which varies the cavity A a and thereby regulates the admission of oxygen through the- tubes B and C into the bomb. Through the centre of the cover but insulated from it passes the wire i which is connected to die platinum wire wj the other end of the latter being connected to e. There is another similar screw valve varying the cavity k, through which -as «] go out of the bomb. To enable the bomb to withstand the osive action of the products of combustion it is plated inside with gold, though platinum would be better. The bomb is enclosed in an HEAT BALANCE IN THE HUMAN BODY calorimeter such as is used for the method of mix! ilorimeter is provided with a and accurate mercury thermo- . i . I he whole is surrounded by n.l temperature jacket. To find the heat capacity of the I. Ir and its accessories a known mnt of electrical energy may be iii in the system or a fuel ol calorific' value burnt. The method is adopted in ing laboratories and the t in actual practice. Benzoic id is most suitable* for this cali- bra t ion. The fue 1, if solid, is formed i ;all briquette; if liquid, it is ul i d in pure cellulose and put in I he platinum dish F and ignited. mi three times the amount of just necessary For complete combustion Is admitted through B, l"' oxygen is generally employ- cd ;n a pressure of about 25 atmos- and at this high pressure the obustion is almost instantaneous, ,i result of these experiments it en found that the calorific i »f anthracite coal, wood (pine), Lrol and methylated spirit are 8.8, Fig. 19— The Calorimetric Bomb-. i.l, 11.3 and 6.4 kilocalories per gram respectively. 43. Heat balance in the human body. — The temperature of the human body remains almost constant in health. The chief loss ol' heat is from the skin and from the excretory functions, the skin niracting in winter to diminish this loss. Heat is supplied to the ly by the food we eat and by the oxidation of living tissues and i ides. The blood stream serves to keep the temperature of the ody uniform. Due to the larger heat loss from the body in winter have to take more food and cover ourselves with heavier clothes. Books Recommended !. Glazebrook, A Dictionary of Applied Physics } Vol. I, article on 'Calorlmei Partington and Shilling, Specific Heats of Gases*. 3. Handbttch der E^perhnenlolphysih, Vol. 8, Part T. CHATTER HI KINETIC THEORY OF MATTER The Nature of Heat 1 Historical,— In the legends of some ancient nations, it is iade for man by some friendly spirit by rubbing -re bis- increase be highly any process the c-ai-lv~philosophers had no correct notion aooui iu ^J p£*lo- "ophS-d I oin the observation that heat could pas, spontaneously from a C body to a cold one. Heat was, thereto-, supposed to be a kind oE fluid— the caloric ftutd. Various fictitious properties were assigned to this JOT*™"" fluid. It was supposed lo possess no weight, since bodies did not rrLe in weigh/ on mere heating. Further, it was supposed u> elastic all-pervading, indestructible and uncreatabk by ' The particle., of this fluid were supposed to repel one another strongly which explained the expansion of bodies when heated d l" o tlielniission of Seat during combustion. Temperature wi , ;,r to potential or level. When the body was heated the caloric id was supposed to stand at a higher level than when cold Pro- of heat bv friction was compared to the oozing out of water ™ m a sp , Th e caloric fluid when thus squeezed out n I itself as i Doi caloric theory of heat began to be thrown cowards Leenth century. The earliest philosopher to have . ,.u:al nature o£ heat was Count RumfonL* In those da a:rc made by casting solid cylindrical preces and inside by a boring machine, Rumford in 1798 hat apparently an inexhaustible amount ot heat could be prod the friction of the spindle o£ the boring machine against body of 'the gun, though the amount of iron scraped was very small lie undertook protracted experiments and found that the amount of heat produced (measured by the raising of water to a high tem- perature) bore no relation to the amount of iron scraped, but was proportional to die amount, of motion lost. He henceforth rejected the caloric theory and asserted that Heat is only a kmd of Motion, Whenever Motion disappears it reappears as Heat and there js an £xact proportionality between the two. He even made an estimate of what we now tali the Mechanical Equivalent of Heat. His value : not much different from the value now adopted as standard. * Count Rumiord (1753-1814) was born in North America Rein r loyal to Grr , .., | i:r [ :T: , the American War of Independence, he had to flee from bis country. He entered into the service of the Prince of Bavaria aiui I in charge of the arsenal at Munich when he perforated the ;.-■:•!, vatec ••-, die bonus of guns. Tn 1799 he went to London and was one of die Sounders of the Royal Institution, Face \h 64, James Pkescott Joule (1818— 1880) (P- fc) Born near Manchester, Joule was educated at home. The main work which occupied the greater part of his life was on the relationship between work and heat. He established the principle of the mechanical equivalent of heat James Cleek Maxwell (1831— Born in Edinburgh, educated at Edinburgh and Cambridge. He became Km* a Colkge, London, in i860 and Professor of Experi- mental P hy5fcs ■ charffe of ^^ ,£££* Abridge, «n 1871. Hi, greatest work was in con- necbon with the development of the kinetic theory of gases and the foundation of the electromagnetic theory of light. (P. 75) FIRST LAW Of THKRMODVNAM1CS 65 In 1799. Davy showed that when two pieces of ice were rubbed gether water is produced. It was admitted by all that water lias greater quantity of heat than ice. I Now supporters of the caloric ; i v asserted that heat is generated in friction because the substance produced by friction has less capacity for heat than the original subs- tance. But the substance produced in Davy's experiment (water) has greater heat capacity than ice, hence the caloric theory became untenable. Davy's experiment proved the greatest stumbling block for the caloric theory. But the valuable work o£ Rumford and Davy was soon forgotten nd it was *■- njy about forty years later I hat the first law of Thermo- dynamics gained general publicity through the researches of Joule in '•land, M and ; Lmholtz in Germany, and Colding in Denmark. 2. Joule's Experiments, — In 1840 J, Joule of Manchester began his classical experiments for determining the relation between the work ne and herd generated. We do not wish to describe these experi- !i detail as they, are at present only of historical interest rhe heal was produced by churning water contained in a cylinder bv means ol brass paddles. This could be kept revolving by means • l a chm I ilc thread wound over a solid cylinder and passing over pulJe ' rying weights at either end. The amount of work i lated by observing the height Lhrough which the weights fell, i ii' rise in temperature was measured by, a mercury thermo- meter and hence the heat generated could be found. After app] '.hi ections Joule found that 772 ft.-lbs. of work at Manchester the temperature of 1 pound of water I°F. In 1878 Joule used ;i modified form of the apparatus in which the work done was ■ the application of an external couple as in Rowland' to be described later (§5.) 3. First Law of Thermodynamics, — The conversion of work into i , thus established by Lhe experiments of Rumford and Joule. e ] the times of Rumford and Joule, the steam-engine n widely applied for various industrial purposes. As we shall see , this is simply a contrivance for the conversion of heat into work. Thus it is established that heat and work are mutually convertible. . ■ may go even further and say that when some work is spent ng heat, a definite relation exists between the work spent 1 n era ted. These two facts, viz., the possibility of converting to heat and vice versa and the existence of a definite relation "ii the two are expressed by the First Law of Thermodynamics. Matin itii illy, the law may be seated thus: — If IV is the work done in generating an amount of heat H, we tii W = JH, I: t This was proved by Black'.? discovery of Latent Heat of Fusion. He equal masses of ice and ice-cold water alternately inside a room and observed that the water was raised 4"C in half-an-ftour, while the ice uiok about ten hours to melt into water, the temperature remaining constant. 5 —7- 66 KINETIC THEORY OF MATTKU. : where / is a constant, provided all the work done is spent in p; i hie heat and no portion is wasted by fricdon, radiation, etc. If H U expressed in calories and W in ergs, J= 4.186 X W- The truth of the second statement embodied in the law is amply proved by the fact that Lhe various methods for finding / (sees. 4—7) yield almost identical values, 4. Methods for determining ./.—Various methods have been d< ed for finding the value of the mechanical q > of heat hut the method of fluid friction and the electrical method are the only ones capable of yielding accurate results, and hence only these will be considered in 'detail. There is, however, an ingenious method of calculating the value of / which was first given by J. R. Mayer in 1843. From the theoretical relation J(G P - C v ) — R (p. 46), he cal- culated the value of /. Thus, Eor hydrogen R = p*VJT» — 8.3 14 X W 1 ergs per mol per °C (pp. 6, 10) and C p -C,= 1.985 cal. per mol per °C (p. 62) . 8*314xlQ 7 .*. J— — -1--985 ~ = 4 - T8 X 1{)7 er S s P er calorie. Certain other methods* that have been employed are enumerated : (I) Measurement of heat produced by compressing a gas— Joule. Heat produced by percussion — Him, (3) Work done by a steam-engii This wi Him in 1862. He measured the amount the cylinder of the steam-engine in a at a known temperature and pressure. The total heal y the engine was found by conducting ill- >• j i>: steam into a calorimeter, and the heat loss due to ci r causes was estimated. Thus the net "••■Lint of heal; which is converted into work is obtai I ik done by the engine was found from an indicator din;; ram (CI ap. IX.). Equating these two Hirn got a value of .7 = 4.18 X 1° 7 ergs P er calorie. (4) Heat developed in a cylinder kept stationary in a rotating magnetic field produced by means of p ilyph: ie alternating I ictric current — Bailie and Ferry, 5. Rowland** Experiments.— Joule's thermometers were not :- dardised and thus errors of 1 or 2% may arise from this cause. Tiie rate of rise of temperature in his experiment was rather slow (about 1 per hour) and hence the radiation correction was : Rowland minimised this source of uncertainty by designing a special apparatus with the object of securing a rapid rate of rise of tempera- ture (40°C per hour) , the principle of the method being identical with that of Joule. ♦ For a ' '.'-.l a$ Llit: methods see Glazebrook, A Dictionary of Api ,. V ■!. I, p. 480. in ROWI -AND S EXPERIM ENTS 67 'I fur calorimeter was firmly attached to a vertical shaft ab [Fig. ch is fixed a wheel kl wound round with a string carrying ;it either end, the whole being suspended by a torsion - A. Fig. 1 (a). — Rowland's Apparatus. wire. The axis of the paddle [Fig. 1 (b) ] passed through the bottom of the calorimeter and was attached to the shaft ef. The latter could >tated uniformly by the wheel g driven by a steam-engine. The number of revolutions jjyas automatically recorded on a chronograph worked by a screw on the shaft ef. The revolution of the paddle at an enormous rate tended to rotate the calorimeter in the same direc- tion on account of fluid friction. This was prevented by the external couple produced by thi ■ <;. p and the torsion wire. For the ' urpose of accurately determining- the radiation correction a water |acket tii surrounded the calorimeter. The paddle is indicated separately in Fig. 1 (c). To a hollow cylindrical axis four rings were attached, each having eight vanes. 68 KINETIC THEORY Or MATTER CHAP. Around these were the fixed vanes, consisting of five rows of ten each, -which were fixed to the calorimeter. Thus the liquid could T£< Fig. 1 (b). F3g.l(tf). b* vigorously churned. The rise of temperature was recorded by a thermometer suspended within the central sieve-like cylinder in which water circulated briskly. If D denotes the diameter of the torsion wheel and mg, mg, the ded, the work W done in » revolutions of the paddle is given by W — couple X angle of fcwist — ffigD- 2/ttfi. . , {*•) the thermal capacity of the calorimeter and its contents, perature i Ldiation) , the heat produced by fri MB8, hence (3) 2mt. tngD 3 — MBd~ If D is in an, and mg in dynes, / conies out in ergs per calorie. Corrections were applied for the torsional couple, for the weights in air which must be reduced to vacuum, for expansion of the til wheel, etc. Rowland found J =4,179 X lu? ergs for the 20° calorie. La by recalculated from Rowland's observations by applying corrections and obtained the value / = 4.187 X 1° T er g s * or ^ 15 ° calorie. Reynolds and Moorby obtained bv a modified apparatus, the value of the mean calorie between 0° and 100 *C to be 4.1833 X 1 T ergs. Hercus and Laby employed what is in principle an induction motor, to find J, and obtained the value / = 4.186 X *0 7 ergs per caloric. 6. A Simple Laboratory Method of finding J.— For laboratory pur- a simple apparatus for finding/ due to G. F. C. Searle is in Fig. 2." but the accuracy, attained by this apparatus is not great. A is a brass cone held "rigidly in position by means of in. ELECT] "R I £ : AL METHODS 69 Fig. 2, — A simple laboratory apparatus for /, non-conducting ebonite pieces attached to a brass cylinder C, which am be made to revolve by means of a motor. Inside A is another ; cone B fitting smoothly into it and attached rigidly to a wooden disc D. The latter has a groove running round its circumfer- ence and carrying a cord which passes over the pulley P and supports a weight mg. When die outer cone rotates rapidly die inner one tends to move in the same direction on account of the friction between the two cones, but is held in position by properly suspending a suitable weight mg at the end of the cord. Tire inner cone B con- tains some water, a thermo- meter and a stirrer (not shown) . When the weight mg ^ is kept stationary the turning moment exerted by it just, bal- ances the frictional couple. If D is the diameter of the disc the frictional couple is mgD/2 and die work done by it in n revolutions of the cone is 2nir.mgD/2. If M be the water equi- valent of the cones and the contents, $0 the rise of temperature pro- duced by friction, then nirmgD — JmlQ, . , - . (4) whence / can be calculated. 7, Electrical Methods.— These methods have already been des- cribed (pp. 37-41) in full detail. It is easily seen that if the specific heat of the liquid be known, this method gives the mechanical equi- valent of heat. There are tw r o methods: — (1) Steady-flow method, (2) Rise in temperature method. The former was employed by Callendar and Barnes who did their experiments with great care and skill but the principal source of uncertainty in their work lies in the value of the electrical units employed. According to Laby we may put the E.M.F. of the Clark cell used by Callendar and Barnes as 1.4335 volts at 1&°C". Tlie*y employed the international ohrn which is equal to 1.0005 X 10° e.m. units. Reducing Calendar's results with the help of these data Laby gets for the mechanical equivalent of 20° calorie a value of 4.1795 X 10 7 ergs. This yields 4.1845 X J0J ergs as the equivalent oE the 15° calorie. The most accurate experi- ments on the subject are those of jaeger and Steinwehr by the rise of temperature method, and of Laby and Hercus by the mechanical method, the respective values being 4.1863 X W a nd 4.1860 X 10T for the 15° calorie, Osborne, Stimson and Ginnings have recently found 70 KINETIC THfcORY Ol? MATTER CHAP, i In: value -1.1858 by the electrical method. Hence, we can adopt the value 4,186 >< 10 r ergs P er calorie. Exercise. — Joule found that 778 li.-lbs. of work can raise the mperature of 1 pound of water IT'. Calculate the mechanical equivalent of heat in C.G.8. units. 778 f L. lbs. =; 778 )< 30.48 )< 453.6 gm. X cm. = 778 X 30.48 X £53.6 X 981 == l-«55 X 10" ergs. Heat produced = 453.6 X $ = 252 calorie.s. ,\ J =z 1.055 X 10i*/252 = 4.187 X 10 T ergs per calorie. 8. Heat as Motion of Molecules. — From these experiments it is [early established that heat is a kind of motion; the- next question is —motion of what ? The answer was given by Clausius and Kronig in 1857 for the case oL gases. They said thai onsists in the motion of molecules or the smallest particles of matter. The idea that if we go on dividing matter we ultimately come to imall particles which cannot be further sub-divided dates from very ancient times, But it remained a barren speculation till Dalton gave to it a definite form in the middle of the last century. The history of the molecular theory is known lo all our read md n< i ount Et need be given here. that according to it all 1 oE a large tiumbei of molecules, all molecules of Line substance being exactly idem regards ■•, etc. ii. En the solid and liquid states these mole- while in gases they are Ear apart from one her. to i lie Kinetic Theory of Matter hear, is supposed to consist in the motion of tin ecules. The identification ol heat with motion of molecules is not a mere hypothesis. It is able to explain and predict natural phenomena and at present there is little i solid foundations of truth. 9. Growth of the Kinetic Theory. — The Kinetic Theory of matter two fundamental hypotheses : (1) the molecular structure ol matter, (2) the identification of heat with molecular motion. The 1 1 'St of these was established early in the 19th century while the second established by the experiments of Rumford, Joule, Mayer and Colding. We may, however, consider Daniel Bernoulli! (1730) as the founder of the modern kinetic theory as he was the first to explain Boyle's law by molecular motions. Clausius and Maxwell in the middle of the 19th century placed the theory on a lirm mathematical basis. Among the other prominent contributors to the theory are Boltzmann, Meyer, Jeans, van der Waals, Lorentz and Lord Rayleigh. Up to the beginning ol the present century, however, the theory had been developed entirely from a mathematical stand-point. There no direct experimental proof of the actual existence of these mole- les or of their motions. Gradually, however, much evidence has •Since the discovery of isotopes tlii: remark requires some modification. in. EVIDENCE OF MOTJttL'LAR AGITATION 71 accumulated in favour of these views, the most important being the Brownian Movement phenomena investigated by Pcrrin in IW&. 10. Evidence of Molecular Agitation.- (1) The phenomena of ssrtsssssse: & nde o C0 2 . Alcohol „.,. d i ffus es into the entire mass or : , water E«n sold is found to diffuse into leaO. k) A jeu tends to expand. It is a common experience that the bh ufcs of the gas 'd To dv away and FT^jZ ansibility. T& rectilinear motion ol the mole- culef is 'ruled by the experiment of Dunoyer in w hi, obtained atomic or molecular y;v t™ annus consists of the tube ABC (Tig. I" u Compartments, A, B, C by p *** ii-mhri«ns 1> and F-. The apparatus is highly e\a- SS2 *fr ic .idc-Bibe J. The end F contain- £ heated to abou Ift^S * through circular apertures in 1 and deposited on . tamd J f ] ; the second hole, form . ' h ^ holes bv straight lines, which proves that the mole- 1 : straight lines. . Phenomena of evaporation and vapour 14')' The m s I aw s ca n 1 >c dedu red from tli e kinetic theory'". . 13). Other results ob from the -, as suecihe heat, iher molecular diameter, etc., agree with experimental results or deductions from other branches of Physics. (5) Phenomena of Brownian movement. 11. Brownian Movement.*- This phenomenon was first discover- cd bv the English botanist brown in 1827 while _ ol aqueous suspensions of fine inanimate spo u der high powei ,pc. He found the spires dancrng about in the wild I, ion. The phenomenon can be readily observed d small particles ided in ; as in a colloidal solution, are examined n , ■ ■ ... __ u A *. n » .. n Jn» 'in iiUr^.mirrosrnTkf. the A F l-\ . " - Dtmoyer's apparatus. ■■■ f, [- detail di ' siom see the Authors" A Treatise or. Heet, |3.4S— 3.4S, 72 KINETIC THKORY OF MATTER CHAP. V/ I Vilhittjih^J" The Browniari movement never ceases -it is eternal and spon- taneous, and is independent of the chemical nature of the suspended panicles, all particles of the same size being equally agitated. Smaller particles are, however, much more vigorously agitated than bigger ones. The motion becomes more vigorous when the temperature is aaed or a less viscous liquid is chosen. It is just perceptible in glycerine and very active in gases. No two particles are found to execute the same motion, hence the motion cannot be due to any convection or eddy currents. Th,e discovery of such spontaneous motion, and the fact that the motion is maintained even in viscous liquids without die application Oi my force was a great puzzle to earlier observers. Gradually, how- ever, it has been established that the Brow ni an movement is due to the impact of the surrounding molecules of die liquid on the Brown jan panicle. It is evident that the Aficroscoj* forces due to molecular impact, will almost completely balance if the size of the particle is very large (say, a large body immersed in the liquid) but there can be no balance if the size is small Any small particle will, therefore, be acted on by a resultant un- balanced force .and will consequently exe- cute motion. As this force varies at random, so the motion of the particle "'.'■ be at random and will be some*' Fi K 6 (p. 81). Thus the phenomenon aent is a d of the existence of w i ion. The study of the kinetic theory is best approached through a ases. The kinetic theory of the liquid and solid states is comparatively undeveloped and will not be discussed m this book. 12. Pressure Exerted by a Perfect Gas.— It has been shown above that a gas consists of molecules in motion. As a consequence, it wre on the walls of its enclosure. To calculate this pressure we first make several simplifying assumptions. These are the following ;. — . (1) Though the molecules are incessantly colliding against one mother and having their velocities altered in direction and magnitude each collision, yet in the steady state the collisions do not affect the molecular density of the gas. The molecules do not collect at one place in larger numbers than at another. Further, in every 1 nient of volume of the gas the molecules are moving in all direc- tions with all possible velocities. The gas is then said to be in a state of molecular chaos, (2) Between two collisions a molecule moves in a straight line with uniform velocity. This is because the molecules are material bodies and must obey the laws of motion. Emufok viug ill. PRiiSSL'RK EXERTED 1SY PERFECT GAS 7$ (3) The dimensions* of a molecule may be neglected in com- parison with the distance traversed by it between two collisions, called its free path. The perfect gas theory- treats the molecules as mere mass-points. 1 1 The time during which a collision lasts is negligible com- pared' with the time required to traverse the free path. (5) JThe molecules arc perfectly elasticf spheres. Further, no appreciable force of attraction or repulsion is exerted by them on one another or on the walls, i.e, ? all energy is kinetic. This is proved by Joule's experiment (p. 47). We now proceed to calculate the pressure exerted by such a gas. We will employ the method of collisions, because it is very simple. Imagine a perfect gas enclosed in a cube of unit sides and con- sider a molecule moving with the resultant velocity c and component velocities «, v, w along OX, OY, OZ axes respectively. " The axes are taken to be parallel to die sides of the cube. The molecule collides with the surface of the cube perpendicular to OX with the velocity u. From the principles of conservation of energy and momentum it follows that it will rebound with the same velocity. Hence the change in momentum suffered by the molecule during collision is %mu. The molecule strikes that particular surface «/2 times per second, hence the change of momentum per second is 2imi.u/2 = ?nu s . Since pressure is equal to change in momentum per second, the total pressure exerted on that surface is "Zmu 2 where the summation extends over all the molecules. /. p x — Xmu*. • (5> Now "'— = w" J where u- is the average of u 2 over all the n molecules. ,\ p K — mnu 2 ..... (6) Similarly, the pressures on the other surfaces are p y = mntJ 2 , p z = mnwK *If we consider the dimensions of the molecule and the forces of attraction, we gel van der Waals' equation (Chapter IV). t The assumption of perfectly elastic collisions, on the average, is warranted by the fact that we can convert into work all the heat supplied tr> a perfect gas. For otherwise addition of heat would increase molecular velocities ami hence also the force of collision, and if deformation of molecule results, all beat may not be converted back to work. The picture here given is essentially that of a monatomjc molecule; there will occur deformations of polyatomic molecules, accompanied by an increase or decrease of rotational and vibrational ci but on the average there is no net los3 or gain of translational energy during collision. Equation (9), however, can be deduced without the assumption of perfectly elastic collision. For details see the Authors' Treatise on Heat, Sec. 3.12, footnote 2. 74 KINETIC THEORY OF MATTER ': CHAP, where a» a , ar 2 denote the mean square velocities in the other two perpendicular directions. Since experiments* show that p x= p^=^&, we have u % z=^iP —~w2. n\ This is also to be expected from the fact that the molecules do not tend to accumulate in any part of the vessel. But n n where c 2 is the resultant mean square velocity. Hence from (7) and (8) IP = £^ and finally (6) yields p = $mnc*. (9) But mn = Ps the density o£ the gas, since ra is the number per ex. p = y <>\ (10) (11) where £ is the kinetic energy per unit volume. Thus we see that the pr. a perfect gas is numerically equal to two-thirds of kinetic energy- of translation per unit volume. ie molecule as if it suffered no collision i others. If a molecule encounters another it gives to ther which bv ;, h as t0 traverfle tance as the first one if there had been no tion (■!) no time is lost in this collision and ur calculation holds true for a perfect <ras. (10) enables us to calculate the' mean squ Kity c" of the molecules of any gas, for 1?=-. $pf p . The pressure ;1 " ; of a gas can be found experimentally and hence c= calculated. Tims the density of nitrogen at 0°C and atmosp! pressure is 000125 gm. per cc Hence for nitrogen the root me square velocity! C = V*- \/- 3 X 76x13-59x981 iM rjm - = 4.98X10* an./sec. *Thk is so for very email cubes. Rigorously speaking, for large cubes tht SSL " "* to k ***** m aC ^ of sSvitation^T ptS on is^mult may b c easily deduced from the laws of elastic impact ? C is the square root of the mean square velocity and differs from the m and 20 i s Ta whale the root mean square of 10 and 20 is /HP +20* _ . V 2*~" =1S ' 81 ' Thi: exact relation between C and c for the molecules of a gas in equilibrium h Riven later (§16), ill. DEDUCTION OF GAS LAWS 75 The formula also shows that the molecules of the lightest gas, ,, hydrogen, would move faster than the molecules of any, other gas under the same conditions. 13. Deduction of Gas Laws— From the above results we proceed to deduce the laws of perfect gases. (1) Avogadro's* Law, — If there are two gases at the same lire p vflft have from (9) p = im i n 1 C t a — sJm 2 n 2 C/., . . . . (12) where the subscripts 1 and 2 refer to the first and the second gas respectively. Further, if the two gases are also at the same tempera- ture we know there will be no transfer of heat (or energy, since the two are equivalent by the First Law of Thermodynamics) from one to the other when ihey are mixed up. On mixing, the two types of molecules will collide against one another and there will be a mutual sharing of energy. Maxwellt showed purely from dynamical con- siderations- that the condition tor no resultant transfer of energy from one type of molecules to the other is that the mean translation a] energy of molecules of the one type is equal to that of the other. Hence if the two gases are at the same temperature it follows that }*&* = &#* (13) Combining (12) and (T3) we get n^=n 2 , (14) I.e., two gases at the same temperature and pressure contain the same number of molecules per c.c. This is Avoga tiro's Law. (2) Boyle's Law. — Equation (10) states that the pressure of a ,;i-. is directly proportional to its density or inversely proportional to its volume. This is Boyle's Law. This holds provided C a remains constant which, as shown above, implies that the temperature remains constant. (3) Dq Hon' s Law. — Tf a number of gases of densities p ti p.p p% ... and having mean square velocities Ci 2 , tV 2 , C 3 2 ....,. be mixed' in the same volume, the resultant pressure p } considering each set of molecules is given by p --- taG 1 % +bA*+WV+' ■ — Pl+P2 + P*+— .♦►♦». (15) ♦Count Amedeo, Avogadro di Quarcgua (1775 — 1856) was born in Turin where he was Professor of Physics from 1833 to 1850. His chief work k Avogadro's law, t Maxwell coiiHidcred the collision of gaseous particles of two different -■ and possessing different amounts of energy. By applying the dynamical laws of impact viz. conservation of momentum and energy, he found that after each collision the difference in energy oi the two molecules diminishes by a certain _ fraction, i.e. the molecule possessing greater energy loses it while that possessing less energy gains, This process is repeated at each collision, and ulti- mately the energies of the two become equal. For details, sec the Authors' Treatise on Heat (1958), pp. 845^847. 76 KINETIC THEORY OF MAI EER CHAP. ITT.] MAXWELL S LAW OF DISTRIBUTTON OF VELOCITIES 77 i.e tf the pressure exerted by the mixture is equal to the sum of the ires exerted separately by its several components. This ia Dalton's law of partial pressures. 14. Introduction of Temperature. — If we consider a grain- molecule ol the gas which occupies a volume V, equation (10) fields pF=z$MC 2 , (16) M being the molecular weight. In order to introduce temperature in the foregoing kinetic considerations of a perfect gas we have to make use of. the experimental law, viz., pV — RT. Hence RT = jMC 2 , x .... (17) fV M Thus C a is proportional to the absolute temperature which may thus be considered proportional to the mean kinetic energy of transla- tion of the molecules. This is the kinetic interpretation of tempera- ture. Hence, according to the kinetic theory, the absolute zero of temperature is the temperature at which the molecules are devoid of all motion. This deduction is, however, not quite justified since the perfect gas state does not hold down to the absolute zero. The interpretation given by thermodynamics is somewhat different (Chap. X) and is more reasonable. That does not necessarily require that all motion should cease at the absolute zero. i put Mz—Nm where m is the mass of a single molecule the iiuniljer oL molecules in a gram -molecule, which is usually called th> □ put R/N = k where k is a constant, k is known as ttoltzrnann's constant. Hence we get p= (N/V)hT = nkT f win: notes the number of molecules per c.c. Further from (17) fcJViiw" - $MT t or W^PT", . (17a> Le. t the mean kinetic energy of translation of one molecule is §&T* Exercise. — Calculate the molecular kinetic energy of 1 gram of heli: : : N.T.P. What will be the energy at I00°C? From (17) the kinetic energy is equal' to Energy at 100°C = 8.5 X 10° X 373/273 ™ 1.16 )< 10*° ergs, 15. Distribution of Velocities. Maxwell's Law. — Tn the above we were concerned only with the mean square velocity and did not care to find the velocity of every molecule. But for studying the properties of the gas further we must know the dynamical state of the whole system. It Is easy to see that all the molecules cannot have the same speed for even if at any instant all the molecules - possess the same speed, collisions at the next moment will augment the velocity of some and diminish that of the others. As the number of molecules is very large (2.7 X !0 ia per c.c. at N.T.P.) and they are too small to be visible even in the ultra microscope, we do not interest ourselves in the behaviour of individual molecules. We treat the Lilem statistically and apply the theory of probability. We shall illustrate this by means of an example,' In a big city there are persons of alleges and we find the number of persons whose ages lie between definite ranges, say between 10 and 15, 15 and 20, and so on. So in an assemblage of molecules where the molecules have all velocities lying between and infinity we find the number of molecules dn t possessing velocities lying between c and c -f- dc. In the steady state this number remains constant and is not modified by collisions. This number is given by die distribution law of Maxwell which states : dn c = 4iraaV**Vi: > .... (18) ere n h the number of molecules per cc, and a = \/bfir = mf%vrkT. But we cannot say what the velocity of any individual molecule selected at random is. We can only say that the probability that its velocity lies between c and c -f- dc is ' Thus the distribution law gives a complete knowledge of the gas only in a statistical sense. A slight transformation ^putting &c s — x 2 ) will show that the number dn = 4wjr~&x*e~^d% t which helps us to represent the law iphically,. Let us plot the function y — £tt~* *""**x 2 against x ti Curve illustrating;: it Maxwell's distribution law Fig. S. — Curve illustrating Maxwell's Law. (Fig, 5) . Then the number dn of molecules whose speed lies between 7b KiNirilO THEORY OF MATTER [CHAP* x and x 4- dx is proportional to the shaded area. The ordinate y gi the fraction of the number of molecules posse- correspond- ing to x, and £ro.m the curve it is obvious that the probability con pon< :; — 1 is greatest, while it is considerably less tor % — 2 or x — \. "Hence we can approximately treaty die whole gas as endov. i with die most probable velocity corresponding to x — 1. 16. Average Velocities.— We must distinguish between two- velocities, the square root C of the mean square velocity, and the. mean velocity c. The former is such that its square is the average of the squares of the velocities of the molecules. Thus 3 UT -■*-±l which we have already obtained in The mean velogit) (17). n dn,.=- tm (20) (21) The most probable velocity a is that value of c for which N e the number of molecules with velocity c is maximum. Hence for such dN = 0. TMs relation gives «■ Substituting get v™=v\ c. An consequence of the large molecular velocities is seen in the almost complete absence of an atmosphere from the surface of the moon. Dynamical investigations show that if a particle is projected from the Earth with a velocity exceeding \/2gr where g is the gravity at the surface of the earth' and r its radius, it wfl never return to the earth and will be lost in space. This critical velocity Is about II Kilometres per second for Earth and 2,1 kilo- metres for the moon. Russian scientists have recently (2nd January, 1959) been able to launch a cosmic rocket "Mechta" which overcame the Earth's gravitational barrier and Hew past the moon space to become the first artificial planet of the sun with an orbital period of 447 days, _ Calculations show that the average velocity of hydrogen at v ordinary temperatures is about 1.8 kilometres per second and accord- ing to Maxwell's law a large number of molecules have velocities much greater and also much less than this. Thus all molecules having than the critical will escape from the planet. Due- certain fraction will always have velocity to molecular collisions greater than the critical and will escape. This loss of the planetary atmosphere will continue indefinitely. It is for this reason that there LAW OF EQUtPARITXON OF ENERGY VI' tically no atmosphere on the surface of the moon while the atmosphere of the Mars is much rarer than that of the earth. 17. Law of Eqnipartition ef Kinetic Energy.— We next proceed ileal with the law of equi partition of energy. It is hetter to> introduce here the idea of degrees" of freedom of a system. Suppose ch an ant constrained to move along a straight line ; it has. then only one degree of freedom and its energy is given by %mx 3 . If , allowecrto move in a plane the energy is given by -hmx 2 4- hny*. An ant cannot have more than two degrees of freedom, but a bee ch is capable of flying has three degrees of freedom, all translation. Thus a material particle, supposed to be a point, can have at most three degrees of freedom. A rigid body can, however, not only move hut also rotate about any axis passing through itself, The most general kind of rotatory motion can be resolved analytical^ into rotations of the body about any three mutually perpendicular a: through a point fixed in itself. Hence the degrees of freedom contri- buted by rotational motion are three. We may now state the defini- tion of- the term 'degrees of freedom'. The total number of inde- cent quantities which must be known Before the position and figuration of any dynamical system can be fully known is called the number :r , : Now it can be shown from rigorous dynamical considerations that nergy corresponding to every degree of freedom is the same- as for any other, i.e., the energy « equally distributed between the various degrees of freedom. This is the law of cquipartition of kinetic energy and was arrived at by Maxwell* in 1859. Boliamannf ended it to the energy of rotation and vibration also. It can further be shown that the energy corresponding to each decree eJ freedom per molecule is XhT%. This law is very general, but we V not attempt here to prove it. Thus if any dynamical system has n decrees of freedom e^fy I tt at T*K is n )< hTtT, ' 18. Molecular and Atomic Energy,— The above theorem is useful in calculating molecular energy of substances. Let. us calculate Luc specific heat of -uses. In a mm gas the molecules are identical with atoms and if, as a Erst approximation, we assume the atom to be structureless point, then from the previous consideration s. each molecule has got three degrees of freedom and will have tlv» kinetic energy equal to &X£hT. In the state of perfect gas the * Maxwell, Collected Works, Vol. 1, p. 378. fLudwig Boltzmaan (1844-1906). Born aud educate! in Vienna, he wi ! esaor of Theoretical Physics at Vienna, G raZ| Munich and Leipzie On account ot his ^ fundamental researches he is regarded as one of the founders of the kinetic theory of gases, mean kinetic energy of $ Equation (17a) gives ±mc a = $kT 1 Le. t the translation per molecule is IkT, If wc assume that the C n< distributed between the three degrees of freedom, the energy associate* with one ,' : °\ t'-.'^om Per mojecule heconins UT. For a formal proof of the see the Authdhs' Treatise oh Heat, §3.26 Ni: KINETIC THEORY OF MATTER CHAP. molecules possess only kinetic energy and no potential energy. The total energy E associated with a gram-molecule is N times the above expression. It is thus equal to %NkT =z$RT. heat at const-ant volume is therefore dE The molar specific C v =\ T - =^\ R = 2 " 98 &d. /degree. For all perfect gases we have established the relation C,~C t - 11. Therefore, for a monatoudc gas and the ratio of die two specific heats J*JL+ Rs!a *±R = 4-96 cal./dcgrce. (23) y =s 5/3 = 1^& These theoretical conclusions agree with experimental results (see i able 5, Chap. II) for the nionatomic gases like argon, helium, etc The specific heat off polyatomic gases can also be obtained by using the equipartition law. The molecule of a diatomic gas may be pictured as a system of two atoms (assumed to be points) joined "dlv to one another like a dumb-bell. The sysi "iti'on to the three components of the velocity of translation of the common centre of gravity, two components of the velocity of out two axes perpendicular to the line of centres of the iras. Thus the system has five degrees of freedom and the energy E = $RT. Hence =$R, G,-1R, y=14. . , . (24) This is approximately the case for hydrogen, nitrogen, etc. (p. 02'). At low temperai • however, C t falls to $R, as Eucken's •riments with hydrogen show, indicating that the rotation hat. disappeared. For chlorine C, is greater than -,R. This shows that Lhe two atoms are not rigidly fixed bur. ran vibrate in a restricted : nner along the line of centres. In a triatomk gas a molecule possesses three translational and •hxee rotational degrees of freedom and hence C „ = fi >< W = SK, C P =4R, 7 =\M,, . (25) more complex molecules y approaches unity but is always greater than it. Tt is not possible to calculate in a simple way the ergy of internal vibration of such molecules since the vibrations are not freely and fully developed. An expression for the specific heat of solid i may also be obtained from the kinetic theory. We can couse ecules of a solid as elastic spheres held in position by Lhe attraction of other molecules and capable of vibrating- in a simple harmonic manner about a mean position. The molecule will have three components of velocity, and nee three degrees of freedom. The kinetic energy associated with each degree of freedom is ^kT. On the average die harmonic vibration i i MEAN FREE PATH PHENOMENA 81 '•, . ergies and hence the total issociated with each degree of freedom is kT. T re, the Eor the three degrees of freedom is $RT f n ! i heat ia 3i£ = .5.955. This yields Dulong and - . I . But the kinetic theory of specific heat is unable to explain the mi o specific heat with temperature (p. 45), particularly, the markctr decrease at extremely low temperatures. Further, the jradual and cannot be explained by the disappearance of se oT freedom which would involve discontinuous changi iplcs of ^R. We cannot assume fractional degrees of fvc the principles of classical dynamics and equipartition law fail lately. The quantum theory of specific heat has been developed whu hi existing facts sai tsfactorily. MEAN FREE PATH PHENOMENA 19. Need for the Assumption thdt Molecules have got a finite Diameter.' — We have seen in the previous secti ms that the molecules i i ;i ■'. iving at ordin . iperatures with very large velocities ; I the case of air it amounts to i ; ; metres per sec. There is no force to restrain the motion of the molecules. Hence the was raised that the assumption of such large rectilinear ies was incompatible with many facts of observation. If the particles in ing with such enormous velocities., the gaseous n :ined in a vessel would disappear in no time. But we ar e that the top of a cloud of smoke- holds together for hours., hence : be .some factor which prevents the free escape of particles. A very simple explanation was offered by Clan sins. He sftoi id that the difficulty disap we ascribe to the molecules a Finite ugh very small volume. Then as :■ i irtirle moves forward, it is to collide witbl another particle after a short interval, ami its tty and direction of motion will be completely changed. The path traversed between two successive collisions will he a straight, line ibed with a constant velocity, since the molecules exert no fori ivei one another except during collision. Hence the path of a single cle will consisi of a series of short zig-zag paths as illustrated : 'i 6. .Some of these paths will be long, others will be short. We < an define a mean free path A . Add I the lengths of a large number paths and divide it by the total number; this will give ■•' - This quantity is of peat importance in tudying a class of phenomena, ca1 port phenomena,, such as vis- cosity, conduction of heat and ion. 20. Calculation of the Mean Free Path* — We shall give a very n :• method or calculating d i mi an free path approximately. We 6 i- 5— Illustration of h kinetic: theory of matter [ CHAP. the one m and .,!:, the simplifying assumption that all molecules except the one under consideration arc at rest. The moving molecule will o with all those molecules whose centres lie within a distance a train its centre U being the diameter of the molecules), and are thus contained in a sphere of radius described about tne centre of the moving- molecule. As the molecule traverses the gas with velocity it wdl collide with all the molecules lying in the region traversed b> its sphere of influence, The space thus traversed in a second is a cylinder ot base W 2 and height v, and hence of volume it<**v. .t the number of molecules per ex. is », this cylinder will enclose ttc-l-. itres of molecules and hence the number v of collisions per second ■ n ffff 2 vnj The length of the mean Iree path V fffl ' 1 111 the above we assumed the other molecules to be at rest. Maxwell corrected this expression by introducing into the foregoing , itions th( motion of all molecules according to Maxwell lisi itioi and obtained the result A— J— W) ■:TTCT~ lumber o isions suffered by a molecule pel 21. Transport Phenomena.— 1 be distribution law expressed b) o be put in the [otto dw, . . : where dn is the number of molecules with velocity romponenl du, v and v -f- do, w and zv -|- dm respectively. 11 nth mass motion represented- by if'-' F*+W*) ■^> du dv <m\ dn = n{, L ,,. I ici-e U = u - U&J v ' '* and W = « - B 11 the gas is not in a steady state any one of the following cases, :• or jointly, may occur. (1) Firstly, u, x , v 9l xti® may not have he same value in all parts of the gas 50 that there will be a relative motion o£ the layers of die gas with respect to one another. We have then the phenomenon of viscosity. (2) Secondly, T may not be the same throughout,, then we have the phenomenon of conduc- tion, viz., heat will pass from regions of higher T to regions of lower T. (3) Thirdly, if n is not the same everywhere we have the case of diffusion, i.e.', molecules diffuse from regions of higher concentra- te regions of lower concentration. It is thus obvious that viscosity, conduction and diffusion represent, respectively the transport of momentum, energy and mass. These are called transport pheno- VISCOSITY 83 ina. These phenomena are brought about by the thermal aeitation the molecules. But the molecules move 'with very lame velo- cities, wink these processes are comparatively very slow. The cause of this anomaly lies in the frequent molecular collisions Hence ■ iiiv of these phenomena is most conveniently done through the chmum of mean free path. The molecules carry with then tarn associated magnitudes and thereby tend to establish equili- 22. Viscosity.-- We shall first discuss the phenomenon of vis- it)'. Here we shall give an elementary treatment of the pheno- nenon based on considerations of menu free path. Consider a gas -i :>n and choose a horizontal plane xy such that there is a mass motion , ot the gas parallel to xv-plane but no mass motion aTong K *axis. Assume that the mass-velocity „ n increases upwards as z ases. The molecules above the plane z = z» possess, on the rage, greater momentum than those below it and hence when xnles from either side cross the plane there is greater transom* tentum downwards since the number of molecules movXsl wy is the same, there being no mass motion parallel to the V-axis We can consider every molecule, on the average, to traverse a equal to the mean free path and then suffer a collision. le velocity gradient is du /dz the difference in the mean molecular oss two planes separated by a distance* A is Xdujdz The 1 » oi a molecule being m , the difference in momentum is mUu^fdz. m. due to heat, motion, the number of molecules moving along the v-ax is must be, on the average, the same as that moving akmg the I the ***«. Hence one-tlnrcl of the molecules may be considered moving along the .axis both up or down, or onlv olS De he oWrvlr m0 T 1Dg 01 ** "P™*" direction. Consider unit area of the observation plane z . The number of molecules crossing this area Si' thf ] " P r rd diimi ? n ** be f ** s ^»ere n /the m, "Jar density and S the mean molecular velocity corresponding to the tperature of the gas, since all those molecules contained in^a cylin der of base. unity and height * cross the unit area in one second Hence the momentum transferred across the plane in the upward 'ion is lnc[ G -mX^) where G is the momentum correspond- to the observation plane. Similarly, for molecules going down- the momentum transferred is ^{c+mX^) . Hence the il momentum transferred downwards ,, -.,;...:. ftp Tllis -„ d z L an accelerating force on the lower layers. Or the lower layers Vtmily we m us t uke the average reSoIvetl pa rf nf , 1 '"• ' ■ out to 1 :• instead of A. Rigorous calculation shows that this will be fytc. along the - ; ■84 y OF MATTER • | retard this fester Layer by a force equal to this. By definition of ;:siu q this force must equal »jd Hence ... (30) e p is the density of the gas. 23. Discussion of the Result— We have established thai but X ==— — - (p. V2 mro* 82) j hence wo* Now is proportional to the square mot of the ab perature, hence the coefficient of viscosity is independci pres- sure or the density of the gas, provided the temperature is co This deduction is 'rather surprising and was first regarded with suspi- cion. Mayer and Maxwell subsequently showed experimentally that lw actually held for prcssm: s lyi *een 760 mm, and 10 mm., unci this was a striking success for the Kinetic Theory. Bin both limits the law fails. At higher pressures deviation from the law ected since the mtermolccular forces ran not be i, At •e path A gradually increases till ii becomes arable with the dimensions of th ' then remains constant Any further decrease of pressure n and pressures the coefficient of \ iscosity ire. Again i) for different gases should vary is found rue. 24. Conduction.— Let us now find an expression for the i. In this case th< rature ies from Iayi i Lo layer and it h the civ which is tran i one lai mother. Considering the dUn the JnfA energy gradient " instead of the momentum gradient m -• I transfer of energy downward? per unit area becomes M_ dz where E denotes the mean molecular energy pertaining to any If K be the conductivity of the gas, -j— the temperature client, the flow ol energ) across unit area in the downward direction is JK , - where / is the mechanical equivalent oT heat. The dT f fE dT j K _ = ^ Xjt _ _ d'E 1 ' ' an — III.] \ ,\U I O] '..NTS when c is expressed in heat units. Further considerations show i above result must be modified into K zeifC v where % is somi and the variation of c- is small, the variation ol conductn Lth pressure and temperature follows in general the tine coi variation of viscosity. Thus conductivity like dependent of pressure. This was verified experimentally in and others. We shall no* consider the phenomenon ot on as ii is somewhat more complicated. 25. Valae of Constants. — The value of the root mean square a gas was calculated numerically for nitrogen on p. 75. lean velocitj c can be calculated from (21) and is equal to C 4.93 X I« 4 = 4,5 X 10* cm. /sec. Then (30) gives, on substitm- 166 X 10- 6 gm. cm* 1 sec-', k- ?*i« I0 " fi -9xl0-«. P c ~ 1-23 :I0-" Mv/IlM ,u cm - i collision* suffered by a molecule per second is Assuming h = 2,7 x 10" per c.c. we have WSTmrAV = ^r':' •! ! y I ,j-e) 5 ! l0_Scm - s will give an idea o\ the numerical magnitudes involved kinetic theory of gases. We give below several constant'; i at a C and atmospheric" | /,./,■' ! moli 10* 1 city in me! per sec. Viis 7X10^ \a gm, cm "' lee" 1 srmal conductivity K X 1C incaL cm. 1 ■src.-i "C" 1 318 52 56 339 38.9 Mean free path Ax 10* cm, Molecu- lar dia- meter 10* cm. C!, 43.1 45 2 493 461 1311 413 30? 166 12 I 9.44 9,95 28.5 10.0 4.5" J. 47 3.5 3.39 218 3.36 4 ■;. , taken from Kayc and Laby f T if Physical and ' . f-ongman. Greet. & Co., 1948. .. at accuracy is claimed values given. 36 K.JUNE11C THEORY Of MATTER Book Recommended. [chap. A good account of the subject matter of this chapter will be found in Eugene Bloch, Kinetic Theory of Gases, English translation published by Me time n and Co. Other Referen/r . 1. Jeai tic Theory of Gases (1940), Camb. IT P. 2. Kennard, Kinetic Theory of Gta 191 raw-Hill. 1. 1 [AFTER [V EQUATIONS OF STATE FOR GASES 1. Deviation from the Perfect Gas Equation.— By the b Quatiorj oJ State" is meant the mathematical formula which the relation between pn volume and temperature of a substance in any state- of aggregation. If any two of these quantiti mown, the third has a fixed value d pending uniquely o and can be determined if the equation of state is known, ill is is seldom possible. According to the laws of Boyle and Charles, we have for a perfect < pV = RT J) This is the Equation for a perfect gas. But even Boyle himself Eound that the law held only under id ons, viz., high temperatures and low pressure, while unaei ordinary conditions it did not correctly represent the true state ol actual gas. Foi every temperature a curve can be drawn which has for its abscissa the volume and for its ordinate the responding pressure of the enclosed substance. Tlr ■■. are led isothermal*, I £ equation (!) wee. isothermals ought to i ingular hyperbolas parallel to each other, but experime 5 ho n Ehe case, The most extended earlier .ttions are due to Regnault. He applied pressures up to 30 heres while the temperature was varied from 0° to 100 G. He duct pV as ordinate ag ; - si a abs< Lssa i ■■■ 6, 7 inira ilii ' I 1 he curves ought to be straight lines parallel to the x-axis; actually, however, they were inclined to it. He foui that for air, nitrogen and carbon dioxide the product pV dei with increasing pressure, while for hydrogen it increases. iU- also found tb n abnormal Joule-Thomson effect (Chap. VI). Thesi facts led him to describe hydrogen as "more than perfect". 1! equation (1) were true the product pV ought, to remain constant; thus these permanent gases were shown to be imperfect. Later work by Natterer, Andrews and Cailletet in 'in- sures confirmed the idea that the actual gases showed consid deviations from equation (1). Andrews' experiments are oJ fundamental importance as they throw much light on the actual iviour of gases and form the basis of an important equation of t'irr proposed by van der Waals. Andrews 1 experiments are ■ibed in the next section. The most thoroughgoing and exact experiments are due to Amagat investigated the behaviour of various gases up to a pressure ol tmospheres. His results particularly" with CO a (Fig, 7) and 1 1 1 . 1 . • 1 1 • showed that their behaviour is very complicated. A different method has been utilized by. K. Onnes who, investi i ihe behaviour of several gases at very low temperatures and that none of the numerous equations of state proposed correctly us the results of experiments. He finds that at any k ■:■.■ 88 i J CIONS OF STATE FOR [CHAP. ANDREWS' JIXPERIM) B9 ture the results are best represented by an empirical equation of the type pVstA+Bp+Cp + Dp^., . , . -. (2) e ^, /J, (7,.., are constants for a fixed temperature, buj vary with temperature in a complicated manner. As many as tw« constants are used; they are called vinal coefficients. A is simply equal to RT while the es of the coefficients uf higher terms diminish rapidly. Holborn and Otto, following Onne* J method, studied several gases up to 100 atmospheres and between the tempera- tures of -183°C and -J- 400*C, and found that they need cake only four cons tits. They give the values of these constants for vai gases at different temp era Lures. The coefficient H •-, •. particular importance. For all gases h varies in l simti; way: at low temperatures it Has a value, then it gradually in- reases tnd becomes positive. Now if at; am peratnre B — ^ and C, D are negligible as usual, then dp Hence, at this -a ture' i.j be obeyed up to pressures. This tem- rnljrd the 2. Andrews' Experiments. — While engaged in ttift attempt to liquefy some of die so-called permanent gases — an important problem of those Andrews,* in 1RG9, was led to study the isothermals of carbon dioxide. His apparatus is indi- cated in Fig. 1, aft is a glass tube whose upper portion con- sists of capillary tube nn.i :'- narrower than the lower part. Carefully dried carbon dias ide passed through the tube for several hours arid then the tube Fig. 1.— Andrews' apparatus. sealed at both ends. The lower end of the tube was immersed under mercury and opened, and some of the gas expelled b 1 heat, so that, on cooling, a small column of mercury rose in the tube and enclosed the experimental gas. ♦Thomas Andrews £18 '-l". ::: :. Born in Belfast, he was Professor oi Chemistry at Queen's College, Belfast, irom 1845 to 1879. He is remembered his work in . with the liquefaction of gases. The tube was surrounded by a strong copper tube A fitted with brass Hanges at either end, to which brass pieces could be attached airtight: with the help of rubber washers. A. screw b ; through die lower flange. The tube A contained water ■•■■ n pressures as high as 400 atmospheres could be app carbon dioxide enclosed above e. To register t ne a similar rair&Wv tube containing; air was placed on the right siue f -m^ in a tube A', exactly similar to A, with which it com- municated through the tube cd and thus the pressures in both tne tub.es were always kepi equal. The pressure in either tube could be varied by means of the screws S or S\ The capillary tube a*- be surrounded by any suitable constant temperature bath (not shown;. 3. Discussion of Results- The curves obtained by Andrews are shown in l''ig. 2. Let us consider die isothermal corresponding tc 13. PC. Starting from the right we see .(portion AB) m>r — that- as we increase the pres- sure, the volume diminishes laiderably and finally lique- i.ll of the gas begins at a pressure of about 49 atmos- pheres (point B) , As lon| u Lctifi ' ' nues the pressure " remains consi.au L and the volume continually diminishes, more mid more of the gas being precipitated as liquid. This is indicated by the nearly horizontal line BC. (The slight inclination indicating an increase of pres- sure towards the end is due to the presence of air as im- purity) . At C all the gas has condensed into liquid and the almost vertical rise: of the curve indicated by (JD corres- i ds to die fact that liquids, nly slightly compressible. The isothermal corresponding to 21.5°C is of the same gen form but the horizontal portion B'C h shorter. In this case the icific volume of the vapour when condensation begins aller le that of the liquid when condensation has completed mi die corresponding volumes lor the previous curve. As tperaturc is raised these" changes proceed in the same direction as above, till at 3L1~'C the horizontal pari 1, - just disappeared and Specific volume in <■£- Fig . nnak for Carbon d ailed the critical two volumes have become identical. This \s isothermal for carbon dioxide. Above this temperature, the horizontal part is absent from all the isothermals and as we increase the: pressure 90 EQUATIONS OF STATE FOR CASES CHAP. SSdlv - riTi ■ i " formatI ? n eL&lMi<L but ,,u: volume diminishes it becomes nearly equal to , ■•!„;„, , ., , ]iqui j at a -Ink lower tcnmey alure . This peculiar! i, o\ ffi isothermal alio ■ disappears at higher temperatures as is evident from the isothermal for 48,1 °C which is much like the isotherm als for air shown separately on the right-hand top. We thus see that the whole diagram for carbon dioxide is divided iiv the critical isothermal into two essentially different regions. Above this isothermal nc liquid state is at ail possible even under the greatest pressure, while below k there are three separate regions. In the region enclosed by the dotted curve BB'PC'C whose highest point P, railed the critical point lies on the critical isothermal, both lid and gaseous states coexist. To the left of the line PC and below the critical isothermal there is the liquid region while to the ■ '. PB there is the gaseous region. Now if, by means of gradual changes, we want to convert gaseous CO.. at 25 C and 60 atm. pres- sure {represented by the point R) into liquid CO. at the same tem- perature (represented by the point S) without any discontinue appearing, i.e., the mass is not to separate into a liquid and a gaseous part with a layer between them, we must avoid reaching the inside of the dotted curve BB'PC'C. Thus we heat the substance above: S3 i and then compress il till the volume he< i to thai of the liquid at that temperature. Next coo] >°C and then reduce the sure. Thus starting from the point r which undoubtedly repre- o/ ike liquik and easeo ,d was [merits of Andre, . bon dioxide is compressed above •i.i f no liquid can make its appearance, however -rem the pressun More accurate experiments show tfrtfc this temperature ; -; 31.0* and not 3T.-1 . The temperature at SI J i is ca led the e (T t i for carbon dioxide. We may define critical tern . ture as the highest temperature at which a gas can be liquefied 1 pressure alone. This is why the earlier attempts to liquefy the permanent gases failed, though enormous pressures amounting to as hi:,! as b,000 atmospheres were sometimes employed. Tli- pressure necessary to liquefy gases aL the critical term is called critical, pressure (p t ) and the volume which the gas* then occupies at the critical temperature and critical pressure is called the me (V,) These three quantities are called the critical mtt of a gas. A table giving the critical constants for various gases, S given on p, 102, It obviously follows that in trying to iquej, -. It uMdw to apply pressure alone if the initial , v : :i :: ;; .. 'v : ' ■ '*■ "* «< ■>,.•, ■■■, ,;, ;1 , M ,,, lhr crisi „, VAN DEE WAALS' EQUATION 91 4 Van der Waals' Equation of State.— 1 he equation of state for i gases was deduced theoretically fiom the kinetic theory of (Chap. Ill) . There we assumed that the molecules have no volume and do not. exert forces of attraction on one another These assumptions are correct only for the ideal or perfect gas to which the actual gases of nature approximate at low pressure and high tempera- ture. Rigorously speaking, however, all gas molecules nave Imite size whose importance was first pointed out by Clausius. tor our pr. purpose we shall treat the molecules as hard elastic spheres. It is clear that at very high pressures the total volume of all the molecules will not be a negligible fraction of the volume of the gas and further even at the highest possible pressure the volume occupied by the substance cannot be less than the volume occupied by the molecules when they are most closely packed. It follows, therefore, that the free volume of the gas to which the Boyle-Charles' law refers is not the *ras volume V but is less than V by a factor b where b is related the total volume of the gas moleci Another simple way of arriving at this result is from considera- tions' of collisions. Consider four balls lying separated from one another on a line perpendicular to the wall and let the farthest one start to move towards the wall with a fixed velocity, if the balls are big, die distance to be traversed is less and hence the last, ball will strike the wall earlier than when the balls are mere points. Similarly for molecules of finite size the number of collisions with the walls and hence the pressure will be greater than for point molecules. Thus the effect of molecular size is equivalent to a reduction in the total volume of the gas by b. Hirn, in 1864, pointed out that the molecules must exert forces of attraction ov- one another, hence the energy cannoL be wholly kinetic, and potential energy due to forces of cohesion must ah taken into account. The correction for Forces of cohesion can be very simply obtained. These fortes are of the same nature as those which give rise to the phenomenon of surface tension in liquids. The mole- cules attract one another with a force which varies ly as some power of the distance between them. Thus die force will be appre- ciable only for small distances and is negligible for larger on: ■•=. molecule in the interior is acted on by forces in all direction's and hence these will balance ; but this is not so with a molecule on the surface or close to it. The components of the forces acting on it resolved parallel to the surface wiM balance but not. those in a perpendicular direction. There will be a resultant force acting perpendicular to the boundary layer and directed inwards. It is obvious that this force on ■ le molecule will be proportional to the number of attracting particles in the fluid, i.e., to the number n of molecules per c.c. The force acting' per unit area of the gaseous boundary will be proportional to the product of the above force and the number of molecules in that area. Hence the cohesive force /?, acting per unit area of the boundary layer of the gas is proportional to ?t~. Now n = N/V where N is the total number of molecules and V the total volume. Therefore />, oc l/V 2 . This force opposes the outward motion of the molecules 92 EQUATIONS OF STATE FOR GASES [chap. aiifl thus decreases their momentum and hence also the pressure generated by their impact. Hence the pressure will be less than that calculated previously by the factor a/pa, where a is .some constant. In other words, we must replace p in die perfect gas equation b} u P + p 1,s 'C the external pressure p on the gas is increased by a/V B . Applying both these corrections the gas equation becomes [*+£)[v-*)-*T. . . . . This is van dcr Waals' equation*. Detailed considerations show that b is equal to four times the total vol tune of the molecules. Van der Waalsf was the flrst to work out a systematic theory, taking into account both these factors, viz., the finite size of molecules and the ces of cohesion. Van der Waals' equation is found to hold n ;i deviations from the perfect gas equation. A comparison of tins with the experimental results is given in sec. 6. Various other equations of state have been proposed. Some of them are more accurate than van der Waals 1 equation in certain regions ; nevertheless, latter, considering its simplicity, gives in general, the most satisfactory first approximation to the behaviour of actual eases. We shall discuss this eqj , re detail. 5. Method of finding the Values of V and V— \ method of - of W and <*> ocairring in van der Waals' equation 1 5 given below : — ' rith-al Data,— In the next n from theoretical considerations that the ■e T„ critical p: l( and critical volume i obeying van der Wa > , vely givt:y • ■ (5) ■ (6) 1 c 27!»R ; Pi H p c Thus j I T * are experimentally determined f a' and 7/ can be I wit* the help of equation (6). It may, however" be pointed out that the method is not very achate u the \-as do t no obey van der Waals' law accurately near the critical pomt. But ot of much consequence as V and <b' themselves defend upon temperature and volume. The values of V at d ' >' Lffev miportanl gases are given in table 1. They refer to 3 rcfof :u. X. I .K, and are determined by this method from equ I ^ on hJS \t ^'a^'f ° f **"** thls «»«*■ "» ** Author,' Treaty tJohatmes Diderik ran der Waals (1837-1923) was born at Lcvdeu m U&t 1)3 Prize in 1910 ^ *"*"* « ^T**** and was Warded < r W)-1S^ : ^t ! ^Jr,^ »*** Y !a the table by D S< , ssi ON OF VAN DTTJl WAALS EQUATION ' ff » 7. — j:\v./r."'.x of 'a' and 7/ /or some gases. . Substance a X W« in atm. )< cm. 4 & x 10* in ex. ium 6.8- 106 Argon 2m 1 X' Oxygen m i Nitrogen . | I 27:1 lV:i Hydrogen 48.7 US Carbon dioa 71V • Ammonia 8BS 1GG ■nes x cm, 4 = 3 -5 x LO 4 . , ise.— Using the values of T e = 5.3, p e — 2.HS atm., calcu- late 'a' and 'b' for helium tor a gram-molecule, 27JP 7 ! ■ x:a-3x ]Q 7 ) a xJ5-3)a ■' - 64 " ,*, ~ 64x2-25x1 '01x10 s atm. x cm, 4 5 3x8 ; 3xl0 7 h ,8xl-01xiO e Discussion of van der equation. — We shall van der Waals' =24 e.e. 6. Waals' now disi i-ion (/>+;; x -« r - This is an equation of the .] degree in V, hence it ows that for every Kalue of p, V must have three values. Further, iroin theory of equations, either all the three values are "real or one is real and two imaginary. Again writing the equa- tion in the form RT a , n . P = yZTl >yT- * (') see that for very large values of V, p is small and in the limit /;=0 when F=Ga. Any in, when V is very small approaching b, p tends to in- finity. Hence the curve must have a concavity upwards. Further V cannot be less than g X ■;,,, i Theoretical curres for carbon divxidz Specific volume Fig. 3— Theoretical curves for CO= accord- ing to van dcr Waals' Equation;, * Taken from Landolt and BurDSlcin, Pkysikaliscfe-Chemische TabcHcv. 94 EQUATIONS OF SI ATE FOR GASKS' ns. — -.*.«— ««g equation by means of graphs lor every temperature (p = ordinate, V = abscissa) curves of die type shown in Fig, 3 are obtained. It is readily seen that tlie turves resemble, in general, the experimental curves (Fig. 2) obtained by Andrews, but if we look for quantitative agreement bv trying to make the two sets of curves coincide we are greatly dis- appointed. In fact, the agreement is only approximate and qualitative party because of errors in the assumed values of V and 'b' and partly, because the equation holds only approximately. There is, however, a remarkable divergence between the theoj iiLiu and the experimental curves in one region. The theoretical curves drawn from van der Waals' equation give maxima and minima in me region represented by straight lines in Andrews' curves. Expert mentally, this u the region where condensation or vaporisation begi and die pressure remain, constant as Ion- as the process continues. 1 his > difference is easily explained when the theoretical curve is. properly interpreted. The part bd inside the dotted curve correspo to the fact thai the volume should decrease with decrease of pre, which a quite contrary to experience. This would be a collapsible state, for any decrease of volume is accompanied bv a decrease of pressure which tends to further decrease the volume. (This is >arent it we imagine the fluid to be confined in a cylinder ') Thus state of the fluid inside the dotted curve represents a state of and consequently, can never he realised in This is why the part bet is not obtained in Andrews' ion ab represents supersaturated vapour and is med experimentally as, Tor exam hen air con- ater vapour is compressed beyond the pofm when condensa- iir, without conden^tion occurring. This ■ from dust or charged ions which ac« T-- This state is, however/ m,,ableandteasik d by the introduction of particles of dust, etc. Hence the portion i M represents supersaturated vapour in unstable equili- brium does not occur ,n Andrews' curves which represent oniylZL of .table exmihbrium. Similarly, the portion de represents a supS hen ted hqmd which is also in unstable equilibrium and is obtained experimentally when gas-free liquid is carefully heated Hcmce is ^nce d re^LT r m An4 ^ ~ Thl ** W-^diver! Van der Waals' theory, however, does not tell us when conden- sation begins, ,.*., where the straight part commences. A Imp e thermodynamic argument* shows that die straight portion shoKe trawn i bat the area abca = area cdec, ' l mi EQUATION 95 maxima and minima points on the theoretical curve can be I b\ putting .^0. Hence from (7) , by differentiation, dp RT 2a _ Q < - (8) (9) 2a(V-b) s RV' A ' i ; cubic equation. Hence, for every isothermal there are i ir one real and two imaginary [joints of maxima or minima. I In- uii vi s below P in Fig. 3 are seen to possess one maxima and one n :i point while those above it have none at all. A slight mathe- nsiormation will show that the other point of minima lies. ion f '<■!.' and hence has no physical meaning. Equation (7) mi. I i'.i when combined yield .' <V • P ~- ~ pa ;io) i ih.; i urve passing' through the maxima and minima points and. n by the clotted curve QPR. In Fig. 2 all isothernials lower than P cut the dotted curve at two points and, therefore, liquefaction can be observed nging the volume along them. For the critical isothermal, 1 1 rraal corresponding to the critical temperature, these His have coah nto one. Referring to Fig. 3, it is readily iai the isothermal passing through the point P, where P is a ! in! lej ion for the family, of curves or a maxima point for the urve, is the critical isothermal because below P every isother- got maxima and minima points while above P there are I all. As we have seen above, if the isothermal has g i. ; ■ linima points there must be liquefaction of the ; i n< I we can find out the position where liquefaction begins. Foi e must be two points on the curve having equal ure. I he highest 'isothermal for which this condition is satisfied tl issitig through die point P since at P the maxima and i points have c >a U seed into one. Hence P in Fig. 3 must be Identified with the critical point and the isothermal through it with tin critical isothermal. Now for p to be the maxima point of the 1 : have, by differentiating (19) with respect to T r and pqu i in- to zero. a_ Sa(Vr-Zb) i (Ifl'i mm! I ' ■mi ■ f/4 V t = 34. b — a P* • 2"b z I = 0, 27m 96 EQUATIONS OF STATE FOR GAS] [chap, r ri? e -^f- C0 T nSta/ be ver y easil V ded «ced from (7) since hi the cntrcal isoiherma] ft* point P is a maximum poin't £ \ ' , a point of inflexion, and for it both §| and |*£ *? e equal to „,. ' have therefore dp _ _ #7* 2n n • •-") 2) •iRIM.F.NlAL STUDY OF EQUATION OF STATE 97 =0, ^bluing (8) and (12) we get p f = 5*, and then with die :,nd %£L=* , 3 ' 7. Defect in van der Waals' EquaUon.-fn spite of agreement deductions from van der Waal* equation show ►le deviation from experimental result,. 1 We haVe r? • shown that for carbon dioxide the curves drawn I SfP. f "■;■'- ^JMkeexp.nmen^ fain van der Waals' equation *ivp$ w — a*, ,..t.i : II h found tha. p, i, Sr^equ^o 2* W "P""" 8 "- I Compression pamp [ he apparatus is shown in Fig. 4. T is the measuring vessel lass which ends in a M h calibrated capillary tube. urt of the vessel is placed iteel cylinder. By means of a V and a screw E the ! is held in the steel cylinder. mostat surrounding the i rt of the tube which losed by means of a mantle .il T is filled with the gas i investigation at atmospheric the space between T and the inner wall 11 being filled with a iiLn ii. in: quantity of mercury over ch a quantity of glycerine or in oil is poured. Pressure is licated from a compression 1 1 through glycerine to the miiig vessel. At high pressures tner< ury will rise up to the capillary ' in and the volume can be easily i r from the calibration. When iIm pressure is high, it is measured by a compression manometer. If the is above 300 atm, and is ted only from the inside the capil- ii l generally smashed, hence Kfc 4.— CajHetct'a apparatus. ih' experimental tube should be subjected to pressure from all sides. uih ;i pressure tube was built by Amagat. Amairat carried out an exhaustive study of the behaviour of several gases by the above method. In one set of experiments he employed pressures up to 450 atm., while in the next series pressures as high as 3,000 atmospheres were used. (b) Apparatus based on the principle of x'anable mass, — This method was employed by Holborn and Schultze. Hoi born and Otto and Kamerlingh Onnes. These investigators worked at high pressures and obtained important results. Since with increasing pressure the volume be- comes smaller, greater error would occur in read- ing the volume at high pressures. To avoid it, these investigators kept the volume constant and used different quantities of the gas whose masses were determined. The apparatus becomes some- what corn.pl i rated by the presence of devices for ntroduction and removal of the gas both inside and outside the 7 Fig. S. — The 1 1 :- sure balance. 98 EQUATIONS OF STATE FOR CASES [ CHAP* experimental vessel- An ingenious pressure balance shown in Fig, 5 was used to measure such high pressures. The metal block B was firmly damped in position and carried the tube T ^frich was con- nected to the apparatus containing the experimental gas. The block B has also a cylindrical hole in which the cylindrical rod R accurately fitted. Between R and the gas in the tube T there was castor oil so that gas pressure, transmitted through the oil, tended to raise the piston R. This was just prevented by the screw S pressing on the top of R with the combined weight, of the frame F and weights W, When balance is obtained, i.e., the piston neither rises nor falls, the gas pressure p z=mgfa, where m is the mass of R, S, F, and "W, and a is the cross-sec Lion of the piston or the cylinder. 9* Discussion of Results.— Amagat represented his results by graphs in which pV de- notes the ordinate and p the abscissa. The curves for hydrogen and nitrogen for several temperatures hown in Fig. 6. As already mentioned, for hy- drogen tire product pi' in- creases with pressure, but for nitrogen it first de- creases. The curves are straight lines inclined to the pressure axis, while if Boyle's law were true, they would be straight lines parallel to the pres- sure axis. The curves for carbon dioxide (Fig. 7) are typical of all gases. The Isothermals 50° have a portion of them parallel to the jbF-axis. This indicates that the pressure remains constant while the volume varies, and corresponds to the condensation of the vapour. Further, it is seen that the curvature of the isothermals diminishes as the tempera- ture rises. The minima points Cm isothermals gradually recede away from the origin, and the dotted curve through them is parabolic. At still higher temperatures no minima point is found and carbon dioxide behaves like hydrogen. This general behaviour of tile isothermals can be easily explained from van der Waals' equation. We have j.OOO S.OOO p In atmos. s,ooo 6 — Amagat's curves (FV again;; for different gases. m In the third and fourth terms which are small, we can make the approximate substitution V = RT/p. Eqn. (IS) then yields SSION OF RESULTS *-*+&-&.-**■ 99 (14) emperatures the term abp' 2 /K 2 T 2 can be neglected. If we then ] us y-coordinate and p as .-^-coordinate, the plot will be a inclined to the pressure axis (e.g. curves for H 2 in too ia& lg3 p in atmos. Fig. 7. — Curves for carbon dioxide. ["bus for temperatures above the Boyle point T B = a/bR will always be positive as for II 2 in the figure. For tem- peratures below the Boyle point as in the case of N 2 and C0 2 in ii and 7, the slope will he negative unless the pressure is too ' • 1 • 1 1 . This can be readily seen from eqn. (14) which holds for the al case, since slope = £^'- +- J*. . (15) ilope is therefore negative at low pressures but becomes positive m| ii iently high pressures. The minima point on the Amagat curves i<\, the point where the slope changes its sign, can be obtained by •qua ting (15) to zero. Thus corresponding to any single value of 1 are two temperatures given by the relation • ■ (16) b ~w = 0, R*T* lotted curve through the minima points is approximately i' 11 1 in Eqn- (16) shows that the dotted curve will meet the axis in she Boyle temperature T B = afbR. Thus Amagat's curves In ltd \ ' •:. ilained with the help of van der Waals' equation. too EQUATIONS OF STATE FOR G [QIAy 10, Experimental Determination of Critical Constants.— We have de- fined the critical constants in sec. $. They are constants characteristic at every substance, and are oi' fundamental importance as they occur in certain equations of state. Their importance in the study of liquefaction is discussed in Chapter VI. The determination of these critical values is often a task of considerable difficulty. Of these Lhe critical temperature is the easiest to measure accurately. For ordinary substances* a hard glass tube- like that of Andrews and connected to a manometer may be employed. Sufficient quantity of the liquid is introduced and the tube surrounded by a thermostat which can be maintained at constant temperatures ring by very small amounts. The temperatures at which the liquid suddenly disappears and reappears are observed, the mean of these giving the critical temperature. The critical pressure is the pressure at the critical temperature and can be read easily from the manometer. The critical volume is much more difficult to measure accurately, for even a small variation of temperature by 0-1° C pro- duces a large change in volume, and hence the substance has to be kept exactly at the critical temperature. The pressure must also be exactly equal to the critical pressure since the compressibility of the substance in this region is very great. The method adopted was to arrange in such a way that a very slight increase of volume low the temperature by a small amount and caused the separation of the into liquid and vapour, the liquid appearing at the top. This initial volume is called the critical volume. The amount of substance initially contained in the tube has thus to be adjusted. The most accurate method, is to make use of the Law of Rectilinear Diameters or mean densities, disco by Cailletet and Mathias. If the density of a liquid and of its saturated vapour be repre- sented by ordi nates and the corresponding 1 temperatures by abscissae, a curve roughly para- bolic in shape 5s obtained (Fig. 8). In the figure the vapour density of nitrogen is plotted from the observations of On- nes and Crommelin and is densities in the two states go on Temperature '. itigradeu Fig. 8.— Law of Rectilineal" Diameter typical of all substances. The approaching" each other till they become equal at the critical tempera- mctimes a simple apparatus, first suggested by Caffniard do la Tour is yea* for the purpose. Il consists of a glass tube shaped like T, w arm somewhat broadened and containing the liquid :n question which is l air in the larger arm by a column of mercury. The two end closed, the air in the longer arm serving as a compression manometer. The ■" :-.r,-ir:cc of the surface of separation between id and its vapour in the sli >rter arm was observed. The curve AB is a line passing through the mean of the vapom i quid densities and will consequently pass through the critical Lempen •. It was first observed by Cailletet and Mathias that for this line was straight or very nearly so. The equation ; y = %(p r \-p n ) = a-[-bt where y is the ordinate and t ■ in abscissa and p u p v denote the densities in the liquid and the lit-; respectively. This law enables us to find the critical ■ the critical volume, for we determine the densities of the ..•apoii and liquid as near to the critical temperature hen draw the rectilinear diameter. The intersect! ivith the ordinate at the critical temperature gives lhe critical iy p c or the critical volume. For substances like water which attack glass at high tempera- la illetet and Colardeau employed the apparatus shown in strong steel tube AB, platinized inside to prevent attack, hi ins the water "or the substance to be investigated. It is immersed im ,i temperature hath LL which is heated by a gas regulated burner, l Fig. 9. — Cailletet and Colardeaifs apparatus. temperature can be kept constant. The tube AB is a similar steel tube FG by means of the flexible steel Mercury fills part of the lube AB, the spiral CDE he tube FG up to the level $ lt above which there is water filling tube up to the manometer. Different pressures can be i! 'I 'I i I by the force pump as indicated, At S t an insulated platinum h is sealed in the side of the wall completes an electric bell n the temperature of the bath is raised the pressure of hi. I thus its ted to ipiia] CDE. EQUATIONS OF STATE FOR GASES [ CHAP. 2 *0- the vapour in OB rises, mercury is forced past Sj in FG and sets the bell ringing. Water is forced in by the pump to keep the level of mercury constantly at S x and thus Uie volume occupied by the water and' water vapour in AE remains constant. The platinum vrirc at Sj completes another electric circuit and serves lo sound a warning that die whole mercury is about to be expelled out of AB. We thus get the vapour pressure curve of the substance. The curve is perfectly continuous and characteristic of the substance. for water this is shown in Fig. 10. If* however, we start with different quantities of the liquid we get the same curve as far as M, but above it we get different curves. The vapour pressure thus appears to branch off at M, a point, whose position was found by Cailletet to be practi- cally independent of the quan- tity 'of liquid taken. This tempe- rature is the critical temperature for the substance, 11, We give below a table of critical constants, taken from Landolt and Bernstein's Physi- cal -Chemisch e Tab el ten. 300 320 3-iO 360 3&0 4 00 Temperature centigrade. Fig. 10.- Vapour Pressure Table Z— Critical data. T c P< Bi RT t Gas in °C. in atm. Specific Pc V* 2-25 volume Helium -267-9 15-4 3-13 Hydrogen -230-9 12-S ;v-\ 2 3-28 Argon -122-9 48-0 1-88 3-43 Oxygen -188-8 49-7 2-S2 3-42 Nitrogen — 147-1 33 -5 3-21 S-42 Carbon -dioxide SI -0 72-8 2-17 3-18 Ammonia 132-2 112-3 4-24 412 Ether 193-8 35-6 3-85 3 • B 1 Sulphur dioxide . - 157-2 77-6 1-95 3 -CO Methvl chloride 148-1 65-9 2-71 3-80 Water 374-2 220 2-6 12. Matter near the Critical Point— There has been much dis- cussion about the slate of matter near the critical point since the time of Andrews. The properties actually observed are: — • (1) the MATTER NEAR TUF. CRITICAL POINT 10! and hence there must be mutual diffusion, and the surface tension must vanish, i.e., the molecular attraction in the liquid and vapour slates must, become equal ; (3) the whole mass presents a very flicker- appearance which suggests that there intent be variations of densitv inside the mass. This was experimentally observed to be so by Hem and others. They suspended spheres of different densities inside the fluid when each comes to rest at a horizontal surface Saving a density equal to its own; (4) compressibility of vapour at rhe critical point is infinite and is very great near that point. As pointed out "by Guoy, this explains the variation of density through- out the mass observed in (3) , for the superincumbent vapour causes the density of lower parts to increase. From these considerations the simplest and probably the most correct view which was put forward by Andrews appears to be that just beyond the critical temperature the whole mass is converted into vapour' consisting of a single constituent and should behave like a gas near its point of liquefaction. According to this theory the critical phenomenon, i.e.., the dis- appearance of the boundary between Che liquid and the vapour and not its motion, should occur only when the amount of liquid in the tube is such that it will fill the whole tube with vapour of critical density. If more liquid is present the meniscus should go on rising till at the critical temperature the whole tube becomes filled with id. If less liquid is present, the meniscus goes on falling till at the critical temperature the whole should become filled with vapour alone. Experimentally, however, Hein found that the critical pheno- menon is observable when the initial density varies from 0-735 to 1 -269 times the critical density. This is probably due to the property (4) as the variation of density inside the mass allows the excess or deficit amount to be adjusted. ' The branching of the vapour pressure curve at M observed by Cailletet and Colardeau may be explained in a similar way. Experiments with water by Callendar point to the existence of a critical region rather than a critical point He found that the ity of the liquid and the vapour did not become equal at the temperature at which the meniscus disappeared, but that a difference of density was perceptible even beyond that temperature. The criti- i 1 1 point is that point at which 'the properties in the two phases line equal. Books Recommended. 1. Jeans, Kinetic Theory of Gates, C. U. P. (1940). Kennard, Kinetic Theory of Gases, (1938), . Glazebrook, A Dictionary of Applied Physics, Vol. I. • CHAPTER V CHANGE OF STATE Fusion — Vaporisation — Sublimation 1. It is a matter of common experience that on the application of heat, »ub&tances change their state o£ aggregation. Thus when ice- is heated it melts into liquid (water) at 0°C (the melting point) . tion point and later the liquid solidifies at the freezing point or solidification point. For a pure substance the melting and the freez- ing points are identical, as are also the boiling and the liquefaction points. The temperature at which any change of state takes place is generally fixed provided the external pressure is fixed and the subs- lance is pure. The fusion point usually varies very little with the pressure (it requires a pressure of about 130 a tin. to lower the melting point of ice to — 1°C), but the variation of the boiling point with pressure is very great. As a matter of fact, it can be easily shown water can be made to boil at any temperature up to 0'-"C.* pro- vided the sufficiently reduced. Conversely, the boiling iderably if the pressure be sufficiently These [acts can be readily verified by considering evapora* sed space. % Evaporation in a Closed Space.— If we fill a glass vessel partly with water and evacuate it with a pump, then water will begin to boil even at room temperature. If the pump be now cut off, the pressure can be measured by a manometer. For a certain definite temperature of the liquid, there is always a definite vapour pressure. Jf we increase the total space, more liquid will evaporate and fill up die extra space. If we reduce the space, .some vapour will condense till the remaining vapour exerts the same pressure. If there is a third gas., noL reacting with the vapour, the partial pressure of the liquid will be approximately equal to the vapour pressure in the absence of the third gas. It. is an important experimental task in Physics to determine "the vapour pressure of a liquid at different temperatures. 3, Latent Heat. — Black found that the change from one state to another is not abrupt, but a large amount of heat must be absorbed before the entire mass is converted from one slate to another at the same temperature.^ Thus to convert 1 gram of * More rigorously, up fcr> the triple point. t Wc have already discussed in Chap. II, pp. 33-37, the methods of measur- ing quantity of heat h>- Change of State »■] CftANGE OF PROPERTIES ON MELTING L05 ice at 0°C to 1 gram of of water at 0°C. about 80 calories of b red The amount of beat required to convert 1 1 gram ot a folidlntoTliquid without raising the temperature ,. called the fa* ioua ™* , .**"i TW amount of heat is required for overcoming the -,, ,, lC mobile enouch to form a liquid. In solids the molecules ^?ima £ne< Ta? "bralin| about mean equilibrium positions which •re fi«d but in liquids they execute rotational ami trans at onal motionflnd wanderVou^hout the liquid, Oiough con»derably * nered Similarly, to convert one gram of water at 100 C to vapour ^1 0-C 538-7 calories are required. This heat which is neces- sary or pulling the molecules of water so far apart that they beo „ qute ndependent of each other (vapour state) r, known as t. Stmt heat of vaporisation. It will be -seen that the latent heal varies greatly with the temperature of vaporisation. A Sublimation,-- Sometimes a solid may pass to a vapour state withLSSh^rfirough the intermediate liquid state. Camphor rrnrles a good example of this class. On being heated it does not ;r. but sSnply evaporates. Such a process is died subhmcUon and the substance is said to be volatile. But we shall see that the process is not peculiar to am parti All solid substances possess finite vapour tension When this vapour tension cular substance. at even ordinary temperature small we take no notice of it, but with the aid o delicate apparatus, ii ran be measured, A substance is said to be volatile only when the boiling point at atmospheric pressure is less than the melting point. Thus under an atmosphere different from our own, say at the moon. even ice which we do not consider volatile would have to be treated as such. The moon is supposed to have a very thin atmosphere t <1 mm. of mercury), and the temperature is below C. If Consider ourselves transported to die moon, our studies will show that ice h volatile because on being healed, it will evaporate to the gaseous state without passing through the liquid state. It we wain water in the moon we must artificially produce a high pressure and applv heat to ice under this pressure. Similarly, camphor can be melted to a liquid form when it is heated under high pressure, 5, Amorphous Solids,— Impure substances, mixtures, and non ■stalling subsLances do not usually have a sharp melting pint; in' their case fusion and solidification take place over a short range of temperature. This is due to the presence of two or more sub- stances which do not solidify at die same temperature^ Examples oi hous substances are wax, pitch, glass etc. Glass gradually tens throughout its bulk as its temperature is raised and is usually irded as a supercooled liquid. 6, Change of Properties on Melting.— Several properties of sub- i ces change in a very marked way when a substance melts. Ous Lo the regular arrangement of' the molecules in the solid being ■ed by the addition of heat. The following are some of these fin iperties :— - 106 l HANOE OF STATE V [ CHAP. (1) Change of Volume, — Most substances expand on solidifica- tion while a few others contract. To the former class belong ice> iron, bismuth, antimony, etc.; paraffin wax and most metals bi to the latter class. Good castings can only be made from substances of the former class. Often enormous force is exerted by water when it freezes into ice. The burs ting of water pipes and of plant cells and the splitting of rocks is due lo this cause, (2) Change of Vapour Pressure. — The vapour pressure abruptly changes at the melting point. The vapour pressure curves of the solid and liquid slates are different and there is a sharp discontinuity at the melting point (Chap, X, § 41). (3) Change of Electrical Resistance* — The electrical resistance of metals undergoes a sudden change on melting. When the sub- stance contracts on melting the conductivity increases and when it expands on melting the conductivity decreases. Table 1* below gives the ratio of the resistance of the fluid metal to that of the solid form at the melting point for a number of metals. Ratio Table 1. resistance of fluid meted rests to n cr o } crystallised metal for some Substan Ratio Substance Cu Ratio Al 1-97 0-70 Li 1-96 2-0 Na 1-34 Cd 1-97 Ag T-9R Cs ->:;. f'.i 0- ■!:■ Ga 0-58 Zn 2-00 An 2-28 Sn 2 - K 1-32 (4) It is also found that molten metals show a discontinuity in their dissolving power at the melting point, 7. Determination of the Melting Point. — The melting point under normal conditions can be determined with very simple apparatus. The substance may be heated In a crucible electrically or otherwise. For high temperatures, the crucible must be of graphite or some other suitable material, and the substance heated in a non-oxidizing atmos- phere. If the substance is rare, it can be employed In the form of a wire (wire-method) . As thermometer, the secondary standards are very convenient to use, the thermo-couple or the resistance * Taken from Handbwh for Fhysik, Vol, X, p, , : 7. DETERMINATION OF THE LATENT HEAT OF FUSION 107 thermometer being generally chosen ; such th ermocouples or resistance wires must not be thrust directly into the melting substance, but should be protected by a sheath of protecting material say porcelain, hard glass or magnesia tubes. Now as long as the substance is melting the temperature remains constant and hence the E.M.I?. of the thermo-couple or the resistance the thermometer is also stationary. curve is plotted with the E.M.F. the resistance as ordinate and time abscissa. The horizontal part re- presents the freezing of the metal and rhp rrvn-psnondintf E-M.F. gives its of A or as the Fig. 1.— Melting point of copper. is about 10-5 mV, whirl; corresponding melting point. Such a curve for copper is shown in Fig. 1 where SQPMORh, Pt couple is used. The constant E.M.F. corresponding to the horizontal part corresponds to 1084°C, 8. Determination of the Latent Heat of Fusion.- For determining the latent heat of fusion, an ordinary calorimetric method (Chap, H) may be employed, e.g.^(l) *e *>«** of ^^ <?. ™ *?"? ice calorimeter, (3) the method of cooling, and (4) electrical methods. (3) is unimportant, and will not he considered here. The method of mixtures.— It is quite simple and has been ex- plained in Chap. II. Most of the early determinations of the latent heat of ice were made by this method. Thus if M grams of ice at 0°C arc added to a calorimeter containing water whose total thermal capacity is W and initial temperature 0i and if <? a be the final tem- perature, the latent heat L is given by the relation ATI+Mff^H^-flJ.- .... (0 An important source of error in the above method lies in the fact that some water adheres to the crystals of ice at 0°C. To eliminate this ice below C C is frequently chosen which requires a knowledge of the specific heat of ice. In this method we are required to find the t taken up by ice in being heated. A converse method may also be employed, viz., the heat given out by water in solidification may found. The most accurate experiments give the value L =zl*d-b i ;if. for the latent heat of fusion of ice. The method of mixtures has been very conveniently adopted to the simultaneous determination of the melting point and the latent heat of fusion of metals and their salts. Goodwin and Kalmus ern- this method for finding the latent heat of fusion of various Salts, A known weight of the substance contained in a sealed plati- m vessel is heated' in an electric furnace to a high accurately i. limbic temperature. It is then dropped into a calorimeter and iiiiLv of heat liberated is determined in the usual manner. The 108 CHANGE OF STATE [ CHAP- 2.— Latent Heat oi Fusion of Salts, electric furnace is specially designed to secure a uniform temperature throughout the platinum cylinder which is measured by a Pt-Rh thermo-couple. As calorimctric liquid, water was employed below 450 °G and aniline above that, temperature. First a blank experiment gives the heat capacity of the platinum vessel. The experiment is then performed with the substance heated to different initial tempera- tures extending over a range of about 50 °C both above and below the melting point, and the final temperature of the calorimeter noted. From these after correcting '* c for the heat capacity of the vessel, the quantity of heat Q necessary to raise 1 gram of the substance from the room temperature to its in ial temperature could be calculated. Plotting Q_ as ordinate and T the corresponding initial temperature as abscissa curves of the type •shown In Fig, 2 are obtained. The discontinuity in the value of Q indicated by the vertical line gives the latent heat of fusion, the tem- perature at which this discontinuity appears is the melting point T m ,' and thi it any temperature gives-'lhe specific heat be substance at that temperature. This methoi rably improved by Awbcry and F. iffiths who have made acci : : termination of the latent heat of iveral metals. employed a very special type of calori- tei which d that the heated substance could be kept surrounded :r inside the calorimeter and the lid of the latter rater hud access to it, so that the loss of liquid by e iti n ed. ctrical Method. — This consists in measuring the amount of electrical energy required to heat a mass of the substance below its melt nt to a temperature above it. The method was employed by Dickinson, Harper and Osborne Cor finding the latent heat of i of ice. Electrical energy was supplied to a special calorimeter similar to that of Nernst in which ice below D C was placed. Then E - J n c pi dT+L- T l n Ct,«dT t where E denotes the electrical energy supplied per gram. 7*^, T 3 its initial and final temperatures and c^ c p ^ the specific beats of ice and water respectively. From this relation L can be calculated. This mediod is very convenient for finding the latent heat at low temperatm Methods similar to the foregoing can also be employed for folding the heat of transformation of one allotropic modification to another •bur we are not concerned with them here. v.] EFFECT OF PRESSURE- ON MF-LTINC POIN 1 109 9. Indirect Method.— Another method of finding the latent heat s-ts in making use of the Clausius-Clapeyron relation (Chap, X) ■Bp (3) where - the specific volumes of the 'liquid and solid respectively, and-^~= the ratio of the change of pressure to the change ol the freezing point. In this way L can be easily found. 10* Empirical Relationships.— It was observed by Richards that if ML denotes the latent heat multiplied by die a torn it weight melting point, ML/ T m is approximately constant for all substances and its value lies between 2 and 3. This generalisation is known as Troulon's Rule* But the relation is only approximately true, and the value of the constant seems to depend upon the nature of the crystalline form in which the substance solidifies. Table 2* i how far this generalisation holds. Table 2.— Illustration of Trouton's Rule. Melting ML Substance Atomic latent point T m ' Crystal system heat, in cab r m Na 030 571 1-7 ) K 570 m 1*7 * Body-centred Rh 520 512 1-7 f cube Cs 590 300 1-7 J Cu 2750 135ft 2-03 j Ag Au 2630 3100 1234 1337 2 IS 2-82 I Face-centred cube Pb 1170 '.■■00 J -W Al 2500 930 2-70 J Mg 1130 927 1-22 } Zn Cd 1800 1500 cm 594 2-60 2-53 / Hexagonal. 560 231 : 2-40 [J II, Effect of Pressure on Melting Point. Regelation— As already mentioned the melting point of a substance is not quite fixed; it I binges when the external pressure is varied. Equation (3) giving »ani!;e in melting point due to pressure has been deduced in Chap. X from thermodynamic considerations. This expression clearly ■ that the melting point of substances which expand on solidifica- II; n i. lowered by increase of pressure while the converse is the case other class of substances, .argcly taken from Hand buck der Experlmentoi-physik, Vol. 8, Part i, CHANGE OF STATE CHAP- Ice belc the first category, it is rfiis of ice which accounts for the well-known phenomenon of Regelation, i.r., the melting o£ ice under pressure and its resolidification when the pressure is released. This property enables us to explain the elegant expert- .i of Tyndall - in which a piece of wire loaded at either end with weights anil placed on a block of ice finds its way through the latter though the latter remains intact. The well-known phenomenon of glacier motion is partly due to the same cause. Snow goes on positing on a mountain and when the mass attains sufficient height the ice at the bottom melts under pressure and begins to flow, bur as soon as the pressure is released it resolidifies. The block of ice thus continuously shifts down the slope and we have the phenomenon • i glacier motion, 12. Fusion of Alloys. — Alloys, except those having composition in the neighbourhood of that of the eutecti* alloy, do not have a j. finite melting point, Consider for example an alloy of lead and i in. The melting of lead is S27°C and that of tin is 232°G, the eutectic alloy having the composition 63^ Sn and &7% Pb. If an alloy of a ana 10% Pb is cooled from the molten state it first becomes pasty at about 210°C when solidiitcation commences and tin begins to separate out, and this continues till a temperature of 1S$°C is reached and the remaining liquid mass which has the composition 08% Sn and 87% l'h solidifies completely. Thus the addition of a little lead to tin or a little tin to lead has the effect of lowering the inciting point of the pure substance just as the addition of a little melting point of ice.' In fan thp behaviour of the alloy is just like that of me salt solution depicted in Fig. 1, Chap. VI. i with a molten alloy rich in lead i.e., 80% Pb the mass first becomes past) at about 275 °C when lead parate out and finally the whole mass solidifies at, 183 C. -s whenever the alloy is very rich in one component, the molten [11 first become pasty on cooling, the paste consisting of crystals held in the liquid, and this will be indicated by a halt in the cooling curve. On further cooling a second halt is reached when the entire mass solidities at 183 4 C< The alloy corresponding to the corn- led the eutectic alloy and this tem- perature of ]iS;l' c C is called the eutectic temperature. Tf we start with the alloy of this composition it will solidify or melt at a definite ii nature — -the eutectic temperature. Similarly if we melt an alloy of composition other than the eutectic, it will first become pasty and then at a higher temperature melt completely. The alloys of Other metals in general behave similarly. Alloys are of considerable practical importance. Thus ordinary soft solder or tinman's solder is an alloy of lead and tin having about 60% Sn i.& a eutectic mixture of lead and tin. It has a melting •John Tyndall (1820—1893), Born in Ireland, ed at Marburg and in Berlin. From IS53 onwards he was Professor of Physics at the Royal Insti- tution in London, He was well-Jcnown as a brilliant experimenter and was i I of ::i::n-i|- VAPOR1SATION i i than Lhat of tin or lead, and has a sharp and nitc |" i .n ure of solidification. An alloy of tin and lead (tin, , i i, at 1J)4 6 C; Rose's fusible metal (tin, 1; lead, 1; at 94°C; Wood's fusible metal (tin, 1; lead, 2: I ; ii LLt ii. 4) melts at G0'5"C though the melting points . and tin are 269, 327, 232°C respectively, 13. Supercooling— It has been stated above that when a liquid i: lilies at a definite temperature (its freezing point). wever, if slowly cooled in a perfectly clean vessel, In down to a temperature much below the normal frcez- ;iit solidifying. This is known as the phenomenon of or surfusion. Water can in this way be cooled down to still lower with a little care. Dufour suspended a minute ma mixture of chloroform and of sweet almonds which • i!i i gravity equal to that of tile water drop and managed latter to — 20°C without solidification while a drop of mplitlialene could be supercooled to 40 °G (normal melting point «l C). soling is, however, essentially an unstable phenomenon. I I I'oduction of the smallest quantity of the solid in which the I pud would freeze at once starts the solidification. Mechanical dis- h as shaking the tube., stirring or rubbing the sides with ■ i.l, or addition of some other solid is often sufficient to start Lion. Tf solidification has once started it will continue with ii ii nl heat till the normal freezing point is reached. After ii Hi'!-' solidification will take place only when heat is lost by i libation, etc. Absence of air favours supercooling probably because [usi particles contained in it are then absent, VAPORISATION 14. Evaporation, Boiling and Superheating.— A substance can pass the liquid state to the vapour state in two ways, viz- (i) evapo- i ebullition or boiling. In the former the formation es place slowly at the surface at, all temperatures, while In the l.ni.'.'i the vapour is Formed in all parts of the liquid a! a • i ii tant temperature and escapes in the form of bubbles producing a Initially these vapour bubbles arc formed around small nibbles of ah clinging to the bottom and sides of the vessel which facilitate the process of boiling. If. however, the liquid is carefully ii dissolved air and then heated in a clean vessel, its tem- i an be raised several decrees above 100°C without its begin- ning to boil, but when boiling starts due to disturbances of any kind,, i ivith explosive violence, usually called bumping, and the temperature falls to 100°C, This superheating is the cause of burnp- i prevented by the addition of porous objects. 15. Saturated and Unsaturated Vapours.- — If the vapour escaping 1 a liquid either by evaporation or by boiling is collected in a it will form an unsaturated vapour. In the case of evapora- in'ii in a closed space as in §2, the space will be filled with un- 112 CHANCE OI- STATE CHAP. saturated vapour if all the liquid has evaporated, but if the vapour remains in contact with some liquid, it will he saturated. Unsaturated vapour approximately obeys Boyle's law but for saturated vapour the pressure depends only on' the temperature and not on the volume, When saturated vapour is heated it becomes unsaturated or super- heated. When saturated vapour is cooled without condensation occurring it becomes supersaturated. We shall now describe some methods for determining the vapour pressure. The range of pressure to be measured varies from 1G -4 mm. to 400 atmospheres. In certain eases, pressures as low at I0-* mm. have to be measured. It is clear that such wide range of values- requires various kinds of apparatus. 16. Vapour Pressure of Water. — The first accurate determination of iTnti pressure of saturated vapour was made by Dal ton. A similar but improved apparatus was later employed by Regnault lor finding the vapour pressure at temperatures lying between 0° and 50°C. Ueg- ria nit's experiments were performed with the greatest care and extend over a wine range of temperatures. His apparatus for the range 0°- 50 c hown in Fig. 3. Two barometer tubes A and B were arranged side side, fed from the same cistern of M, The space a above the mercury level in the tube B is- vacuum, while water is gradually introduced at the bottom of A till il rises through the mercury column and evaporates on reaching h. More water is introduced till a small layer remains floating over the mercury surface in A. A constant tempera- ture bath DD furnished with stirrer (not shown) , and a thermometer surrounds a, b f as well as some length of the mercury column. The dilTerence in the heights of the two mercury columns, which ;<• _ observed through a glass window with a catheto meter, gives the saturated vapour pressure of water at the temperature of the hath. Correction must he made for the weight of water in A, for effects of capillarity, refraction, etc. For re trip: ratines below C C "Rcgnault modified his apparatus to ol Gav l.ussac. The top of the tube A was bent round and • a spherical bulb which contained water or ire and "was surrounded by a suitable bath. For temperatures not much above 50 C 'C the apparatus already described (Fig. §) could bt: used when a Fig. 3.— Regnauli • — " ;al methods 113 few C i; loijfl ith would necessary, but Regnault preferred the boiling tec. 18) . 17. General Methods.— The methods used for measuring saturat- i sure can be broadly divided into two classes • in which the temperature is kept fixed and the is determined either manomelxically or by measuring tire 1 s iturated vapour. This is called the direct or static method. : Metliods in which the pressure is given and the temperature ii It die liquid begins to boil is determined. This is the dynamic 18. Statical Methods. — Regnault's method is illustrative of class i 1 ) The same method can be adopted For finding the vapour pressure ol any liquid provided it ot react with mercury and the vapour pressure is neither too high nor too low. A number of investigate employed this method. Their apparatuses differ only in un- iitiai details. A general scheme of apparatus utilising this i md is shown in Fig. 4. A is a small glass sphere, a - i lacity, to which is attached a glass tube incl another glass tube D with a smaller bore. I In; is connected to a bigger globe G and a ii' n ury manometer M about 90 cm. long. The whole apparatus is first evacuated through the itop cock Sj. and then the latter is closed.' Next the gas under investigation is introduced the stop-cock S 2 and condensed in A by cooling the latter; finally S 2 is also The sphere A is then surrounded by i] in me baths and the vapour pressure col- liding to the temperature of the bath is a ted by the manometer M. B is a baro- to indicate the atmospheric pressure. The apparatus is convenient for measuring vapour tires from a few cm. to the atmospheric For higher pressures a compressed air ii 'meter may be employed when the whole as to be made of steel, \ similar apparatus was employed by Hen- I Stock for finding the vapour press > ol a number of gases between -f-10 and On the same principle Siemens has I I vapour pressure thermometry at low 1 ' For high pressures we may ii the classical experiments of Cailletet "il' n (p. 101) with water, Andrews' Apparatus (p. 88) may also be used, lluiborn ind 13a ami determined the vapour pressure "' watei ahovi 200 a C by the statical method, [he methi also been employed by Smith and Menzies though i] i is is ranch different. .A Fig:. 4.— Determination of vapour pressure by statical method. 1 114 CHANGE OF STATIC CHAP. Ill the experiment mentioned above the pressure was measured manorjaetrically, but Uie pressure may also be found by determining the density of saturated vapour, for assuming that the perfect, gas equation holds, we have pST P^' M <*) Thus knowing M, p we can calculate p. A very simple apparatus based on this method was employed by Isnaidi and Gans. A portion of the vapour was isolated and its density determined. 19, Dynamical or Boiling Point Method,— This method is based on the fact that when a liquid boils, its vapour pressure equals the external pressure on the surface of the liquid, A definite exter- nal pressure is applied on the liquid surface by means of a pump and then the liquid is heated. The liquid will 'boil at the tempi ture at which its vapour pressure equals tile external pressure, that, set up by the pump. Hence the vapour pressure corresponding to the temperature of ebullition is the pressure exerted by the pump and can be read on a manometer. Fig. S» — Iiegnault's Vapour Pressure Apparatus (Dynamical method.) Regnault employed this method for finding the vapour pressure of water betwe 50°C and 200 D C His apparatus is indicated in Fig. ft. The copper boiler A is partly rilled with the experimental liquid and contains four thermometers be immersed to different depths, inside the vapour and the liquid. The upper part of the boiler is BOILING POINT METHOD 115 cted by means of H to a pump, tfie pressure being indicated bv n i cury manometer NM. The reservoir is kept immersed in a bath VI and serves to transmit the pressure from the boiler to ' manometer as well a, to smooth the fluctuations in the pressure til ined by the pump. The vapour of the liquid condenses in C uul returns to the boiler, thus the same quantity of liquid is used over and over again. First, a definite pressure is established by the pump and the boiler heated. In a short Lime the readings indicated bj i lie thermometers be become steady. The manometer indicates trie vapour pressure corresponding- to this temperature. The apparatus can be adapted [or all pressures. For high pressure ■.'• ihe parts must be made of copper and the pump must be a force SL,^^ is capable of great accuracy/ By this method >rn and Henmng have very accurately determined the vapour -sure of water between 50 and 200°G ft to 16 atm.) . Ramsay aid ug app^ed this method to the measurement of very small vapour ■■urts. J heir apparatus (Fig. 6) consists of a wide glass tube i to which is connected a reser- • R, the latter being connect- to an air pump and a mano- meter M. Definite pressure is '.<i up by the pump and the ex- ttal liquid, stored in F, llowed to drop on the cotton • 1 surrounding the bulb of the thermometer. The tube T kept surrounded bv a suffi- iy hot bath so that the ex- perimental liquid aL once evaporates inside T and the thermometer soon reaches a steady temperature, the true boil. mg point ot the liquid under that pressure. Smith and Memies have devised an inireni. ous modification of the boiling point method Ineir apparatus is indicated in Fig, 7. The sub- stance under investigation is kept in the sphere A close to the bulb or the thermometer T, both "ttng immersed in some liquid contained in the ^-tube 15. . The test-tube is dosed airtight and communicates with a pump and manometer t shown) . It is further surrounded by a bath whose temperature can be varied. A 'definite pressure is first established by the pump and the temperature of the bath gradually raised. When the vapour pressure of the substance contained m A becomes equal to the external pressure on the surface of the liquid in the tube, any further increase of temperature increases the pressure of tire vapour m A which consequently bubble* w Fig. 6.— Ramsav and Young's apparatus. JJc=* '''" i and apparatus. - 116 CHANGE OF S' [ CHAP, \ . LATENT HEAT OF VAPORIZATION 117 through the liquid in the tube. When this just happens the press un recoX by the manometer ^ the vapour pressure corresponding to the temperature indicated by the thermometer. The boiling-point method has also been applied to metals by £e .."cadmium was heated electrically in a quartz or por, Jeltin tube and the temperature of the vapour was measured by a thermo-couple. 20. Discussion of Result-Experiments show that the saturated vapour pressure of every substance increases as the temperature n raised. Hence the vapour pressure f> must be a function oi the tem- perature viz ft — f IT) where / (T) must be of such form that its % aluc E522 Sh "the' Utperature T, Various empirical relation*™ between ft and T have been proposed from time bo time. They hold for limited ranges and are by no means quite exact, and universal. In 1820 Young proposed a very simple formula log p = A + y> and are constants. where T is the absolute temperature ana A, KirchhoR in l I Rankine in 1866 proposed quite independent^ the form li la B \tygp =A+ r -fClogT. m % This formula agrees with experimental results very closely and can also i„. il considerations. It is shown in Chai that dp_ a) e fs h the vapour pressure. L tbe latent heat of vaporization, T loint and v* v t , the specific volumes of the substance in the liquid and vapour states. Neglecting v x in comparison with v s v 2 by R T/Mp from the gas laws we get . - • (8) lo g p= R \ Y ,-+u Tf. Assuming £ constant, equation (8) yields Young's formula (5) . •::v-£ v. , we assume L to vary linearly with temperature, i.e., L = L n — i*Tj, equation {9) yields us formula (6) . In order to get the exact, value r -l' pressure corresponding to any temperature we must use an accurate expression giving the v due of L as a function of T, The above holds for the saturated vapour pressure of a pure liquid; when mixtures of two liquids are investigated they yield bi- ting results, (1) Tn case of liquids which do not at all mix the :. ii- pressure of the mixture is equal to the sum of the vapour pres- sure of the constituents, e,g*, water and benzene or water and carbon (2) Iii case the liquids are partially miscible the vapour in e is always less than the sum of the vapour pressures of the con- s and may even be less than that of one of them, e,g., water .iii.l ei tier, or water and isobutyl alcohol. In this case each constituent Little of the other and the vapour pressure of the mixture aains constant over a large range of composition of the mixture but falls off for very dilute solutions finally attaining the limiting- value for the pure constituent. Thus if we distil a weak solution of isobutyl alcohol, we find that the solution in the still becomes weaker and I finally there is only pure water in the still and pure alcohol in i he condenser! For a weak solution of water in alcohol the reverse ase. (3) When the two licjuids are wholly miscible, e.g., water and methyl alcohol, the pressure is intermediate between those of the arate constituents. In this case whatever composition we start with, alcohol always passes to the condenser leaving pure water in the still. 21. Vapour Pressure over Curved Surfaces. — In the foregoing we have considered the vapour pressure over a flat surface. The vapour pressure over a curved surface is different on account of surface ten- i. Evaporation from a spherical drop produces a decrease in area and hence also in the surface energy due to surface tension and therefore, it will proceed further than in the case of a flat surface, i.e., the vapour pressure over a convex surface will be greater than that over a fiat surface. Detailed considerations yield the result taA.^1 (9) re fa, p denote the vapour pressure over flat and curved surfaces respectively, S the surface tension* r the radius of curvature of the rface (considered positive for concave and negative for convex) and p tjie density of the liquid. These considerations have important nsequenr.es in the precipitation of rain and in the phenomenon of []i v. Equation (9) shows that if r is small and n* Le. 3 rface is convex, p may become very large. Hence, if suitable nuclei for condensation are absent a high degree of supersaturation ■ he attained and inspite of it, no drops will be formed, LATENT HEAT OF VAPORIZATION 22. Tn the- measurement of latent heat of vaporization we have i" remember two points; first, that k?\ absolute value is relatively 'y, secondly, that the latent heat is absorbed or evolved in an isothermal change of state. The consequence is that its experimental deterj lination is very little affected by the usual sources of error which isent in all ealorimetxic measurements. The methods can be grouped under three broad headings: l) Condensation Methods, — Those in which the amount of I it ".ii evolved when a certain amount of vapour condenses h measured, (B) Evaporation Methods. — Those in which the amount of heat requ i vaporize a given mass of the liquid is measured directly. "1 118 CHANGE OF STATE [ CHAT*. v-| EVAPORATION ME1 I 119 The heat is generally added in the form of electrical energy and can 1 easily determined. (G) Indirect Methods. — Those in which the la Lent heat is cal- culated with the help of some thermo dynamical relationship such as Clan GJapeyron relation (equation 7) . From the vapour pressure curve the quantity dp/dT is determined and hence L evaluated from (7). We shall now con- sider the first two methods in greater detail A. Condensation Methods 23, Berthelofs Apparatus, — -Reg- nauit's experiments are rather of histo- rical interest We describe below the apparatus ol' Berth Hot (Fig- 8) , In this apparatus which is wholly of glass the liquid is kept in the vessel D and heated by the ring burner B, The rest of the apparatus is protected by an insula mantle M, The evaporated gas passes through the tube T into the" spnal S, placed within the calorimeter |C. The iral S is fitted to T by a conical ground pio an be easily remo 1 The vapour condenses within the its latent heat lorimeter which can be easily measured by observing the rise of temperature on die thermometer placed inside a jacket. The amount of water condensed is obtained by welch- ing the spiral S before and after the experiment The heat, measured the heat of vaporization plus the heat given by the con- I liquid in cooling- from the boiling point to die final tempera- ture of the calorimeter. The open end or S is connected to a pump to regulate the pressure under which the boiling takes place. Errors are likely to arise owing (1) to superheating of the liquid, (2) to minute drops of wai ing carried over by the vapour. As the: use of a ring burner causes the heating to be sometimes irregular, Kahlenberg replaced the ring burner by a metallic spiral placed inside the liquid and heated electrically. 24, Awbery and Griffiths used a slightly .modified apparatus in which the usual calorimeter was replaced by a continuous flow calori- meter. The apparatus is shown m Fig. 9. The boiling chamber is heated electrically by an inner coil. The vapour passes down die vertical tube which is surrounded by a jacket of water through which a stream of water flows at a constant rate. The temperatures of the inflowing and outflowing water are determined by two thermo-couples. There is a third thermo-couple at the mouth of the vertical tube which L— Berthelot's . • LUS. \ Ijjpti htasins «fl iij?3ic&Gntei>i} gives the temperature of the condensed liquid as it leaves the appnra- In this experiment the . i pour must be produced at a stead 1 .' rate and this is achieved by the use of elec- trical heating. The latent at is obtained from the i" mula M8=m[L-t*(t s —t l )l (10) where 9 is the excess of temperature of the outflow water over the inflow water, M is the quantity of water flowing per unit time, m is the rate at which the liquid is being distilled, t 2 is the boiling point of the liquid, ,*-, the temperature of the Ii- td as it leaves the appara- tus. B. Evaporation Methods 25. This method wa i employed by Dieter id. who measured the heat required to evaporate a given mass of water with the help of a fiunsen ice-calorimeter. The water was contained in the tube A "of the ice- calorimeter (Fig. 3, p. 34) and the heat was measured by finding the mass of mercury expelled. Griffiths found the electrical energy required to vaporize a given mass of water. We shall describe the apparatus used by Henning For precision measurement of the heat of vaporisation between 30 p and 100°C. 26. Henning's Experiments. — The apparatus employed by Hen- ning- is indicated in Fig. 10. C is a copper Vessel, one litre in capacity, in which the liquid is allowed to evaporate. This is surrounded by an oil-bath A maintained at a constant temperature. The heating ! place through the spiral D of constantan wire wound on a quadrilateral mica frame. E is a platinum resistance thermometer. Tfu- vapour which is evolved passes through the German silver tube H downwards through the German silver tube KK to the vessel P in which it is condensed and weighed. The end of H is bent down- wards so that no liquid drops can be carried. The vapour is first ! Is P, and when the conditions become steady h i op-cock R h turned so that steam is led to the other vessel P. tfter a sufficient quantity of steam has been led to P the stop-cock turned to the other side. Fig. 9.— Awbery and Griffiths* Latent Heat Apparatus. 120 CHAKG£ OF STAT: [ CHAP- The quantity of vapour deposited in the second vessel is now Lou id b) weighing and the quantity of heat ' supplied " is obtained from observations of the electrical measur- ing- apparatus. For determining; the heat of evaporation at lower pres- sure, P is connected to a large vessel i i about 5 litres capacity which is maintained by means of a water pump at the required pressure, A similar apparatus was em- ployed by Henniiig Tor finding the heat of vaporization of water up u> I80°C when the pressure reaches about 10 atmospheres. Fogler and Rodebush used this method for deter- mining the latent heat of evaporation of mercury up to 200 For determining the Intent heal of evaporation of substances like nitrogen, hydrogen, helium, etc. r which become liquid at very low temperatures, the above principle has been utilized by #Dana and • r . . t_r * Ormcs, Simon and Lance and _ieat ° others. DISCUSSION OF LATENT HE J DATA 27. Variation of the Latent Heat with Temperature. — Experimenl ; : ::j latent heat diminishes as the temperature at which boiling takes place is raised. This was no v&a by early ors who proposed various empirical foruuilae. Of these Thiesen's formula appears to have been most satisfactory and states L^L X (*,-!)* - (11) where t t is the critical tempe- rature and Lj a constant winch tfces the value of L at t = t & - 1, This is of course based on the assumption thai latent heat vanishes at the critical temperature, Henning showed that between 30° and 100-C the latent heat of vapo- rization of water is given by the formula L= 538.86+0-5994(100-0. (12) Fiff, 11,— Variation of Latent Heat of CO j with temperature. trouton's rule 121 ._itrgator. shows dearly that the latent heat vanishes at the critical tempera. e of CO, which is about SPG This result is quite, universal. e exact variation of latent heat with temperature for all substances iven bv the thermodynamic formula where c e) c s denote Che specific heat of the substance in the gaseous and liquid states respectively and v it v t , the respective specific volumes, 28. Troirton's Rule*— As in the case of latent hear of fusion, we have here also an important generalization known as Trouton's Rule which states that the ratio of the molar latent heat arm- the boiling point is a constant for most substances; or symbolically, &L= constant, .... (14) where M is the molecular weight and Tt, the boiling point. The ue of the constant is about 21. The law does not hold for asso- rted vapours. Table 3 shows that tire law holds approximately for mc;;: substances, Table 3 — Illustration of Trouton's Ride. Substance Gra m-mol ecul ar latent heat in en lories Ml ii Pin- point T, v 4-29 20-4 77-3 96-1 188-1 859-5 809-0 319 353 :■:■ 680-0 858 942 1155 1180 1887 v , r ML Value O: ,. 1 b experimentally Helium Hvdrogen Nitrogen Oxygen ochlonc acid Chlorine : anc Carbon disulphide Benzene Aniline Mercury slum "Rubidium Sodium Zir.c Lead 22 219 1310 1030 3890 4600 6100 6490 7350 10000 14200 15600 18700 29300 27730 4f)000 5-1 10 17-3 18-1 20-7 . 19-75 20-8 21-0 22-6 18-2 19-9 20-2 23-5 24-4 T 122 CHANGE OF STATE [ CHAP, V. DENSITY OF SATURATED VAPOUR 123 It will he seen, however, that ML/T is really not quite constant for all substances, A simple theoretical discussion shows that it can- not be so., for the boiling point under atmospheric pressure is purely an artificial point and has no physical significance. It varies enor- mously with the external pressure, while the latent heat, varies neither in the same direction nor to the same extent. In the case of water the Trouton quotient at a few temperatures is given in Table ■:. Table 4. — Trouton quotient for water at nt tempera? urex. Pressure 4-6 mm. 760 latent heat Trouton quotient 606-5 cal. ; 40 25-9 16 From the variation in L and the boiling- point it is evident that the Trouton quotient will go on decreasing as the temperature becoming zero at the critical temperature where the latent heal rushes. Thus the quotient can have any value from. to 40 and it appears to be merely an accident dim • t substances boiling undi the value is about 20, # DETERMINATION OF VAPOUR DENSITY 29, sity we mean the spec the vapour lir or hv ity. The vapour density can be ea known weight of vapour ai a certain pressure and temperature. Thus if to grams of die v c.c. at the pressure p and temperature j", and p rt it of .. of the standard substance (air) at 27i' IJ and 760 mm, then the vapour density is given by w T_ 760 /V 273 'p W The density of the unsaturated vapour can be easily and a ecu rately found by any <.uiq of the standards methods* viz. ? of Victor Meyer, Dumas, lio'fmann and others. But before I860 there was no method for directly determining the density of saturated vapours. The methods adopted were all indirect in which the density of the unsaturated vapour was first determined, and assuming the perfect gas laws to hold up to the saturate:..] state, the density of' the saturated vapour was determined. This assumption is, however, not quite jus- tifiable, hence these methods can never give accurately the density of iturated vapour. Still, however, they are frequently' used especially *Fufl details of these methods will be found in any text-book on Physical Uxemistry. dial of Victor Meyer, which is of considerable practical importance is consequently described below. The apparatus (Fig, 12) consists of a cylindrical bulb B with a long narrow stem, ttear the top of which there is a side-tube. The lower part of the tube is surrounded i suitable temperature bath which is .' t epl nished by a suitable liquid boil- ing at some pressure in A. Air inside the tube gets heated and is expelled at the top; after a time, however, a steady state is attained when no more air escapes. The substance whose vapour density is to be determined is enclosed in a thin-walled stoppered bottle and placed inside the tube. By manipulating- s it is allowed to : gently in 15. The bottle breaks, the liquid vaporizes and thereby displaces an equal volume of air which escapes at the side-tube and is collected in the tube L Knowing the mass of this air, rhe density of fl r of the substance is obtained the mass of the liquid taken by the mass of displaced air. rnst has modified the apparatus and could thereby measure the vapour density of KC1 and NaCl up to 2000°C while Wartenberg found the vapour density of several metals up to 200f.PC, Foe a description see Arndt Pkyxifialisk-chernische Tech' mk. Chap. IX. 30. Accurate determination of the Density of Saturated Vapour* — In 1860 Fairbairn and Tate sed an apparatus by means of which they mea- sured the density of saturated vapour directly. Fig. IB explains the principle of their apparatus, A is a spherical glass bulb whose narrow stem dips into mercury contained in the outer wider glass tube. I r communicates with the metal reservoir B. Both A and B contain some water a hove the mercury levels, the latter containing a larger quantity than the former. All air is expelled from the apparatus and then both the vessels are surrounded by a bath whose temperature is gradually raised. The levels of the mercury in the two vessels remain con. that in A being- always higher than in B due to the excess of water in B. This is so as long as there i% any liquid water in A. But as soon as the liquid in A disappears, the level of mercury in A suddenly rises. This is because the saturated vapour pressure increases much faster than the Fjjt. 12.— Victor Mej Vapour Density Apparatus- Fig, 1&— Fair- bairn and Tate's apparatus 124 CHANGE OF STATE CHAP. I pressure of unsaturated vapour obeying Boyle's law. The tempera- ture at which this sudden rise of mercury column in A appears is noted. At this temperature the vessel A becomes filled with saturated or whose pressure may be found by means of a gauge connected to B. Knowing the mass of water in A and the volume of the enclosed space, ihe density of the vapour at the particular tempera- ture and pressure can be calculated. Somewhat later Perot attempted to isolate a portion of the saturated vapour and to weigh it. This could be very conveniently done by isolating the vapour by means of a stop-cock and getting it absorbed in dry calcium chloride and weighing the latter. K. Qnnes employed another simple method. In a graduated vacuous tube different masses of the liquid are introduced and the volumes of the mr and the liquid observed. Thus if m and m' grams of the substaj introduced and the volumes occupied by the gas and the liquid are Vi and v» in the first case, and v\', v? in the .second case, and p g , p t represent the densities of the gas and die liquid, we have whence •—-'•■;- i , «'=*»!>* +P»'W, "r.'^- r 'i. "t (16) Book Recommended. azebrook, of Applied Physics* Vol. 1 1 article CHAPTER VI PRODUCTION OF LOW TEMPERATURES I7e low^nP^wVc obtainable. The study of the laws o per SrC can, at least theoretically, proceed on the «n- £5fe*te SJ degree, below the ^eUing point of rcc ; a, ;1 £. i?.he lowest temperature conceivable. This is taken as the zeio of tne aSte^mperaUire scale. In this chapter we shall dtscu^ die principles and contrivances by which die region from C to absolute zero can be reached. PRINCIPLES USED IN REFRIGERATION 2. For reaching low temperatures we have to utilize proc, ; bv which a bodv can be deprived of its total heat content. Ihe tol- lowing methods' may be employed to achieve this end :— (i) Bv adding a salt to ice. m Bv boiling a liquid under reduced pressure. hit) By the adiabaiic expansion of a gas doing external work. (iv) By utilizing the cooling due to Joule-Thomson effect. (v) By utilizing the cooling due to Peltier effect, (wi) By utilizing the heat of adsorption. (vit) By the process of adiabatic demagnetisation, A general theory of refrigeration will be given later in Chap, IX; here we shall simply discuss the principles and contrivances for utilizing them. (i) Adding a Salt to Ice 3 Low temperatures may be attained by adding a salt to Ice This is the same process which was employed by Fahrenheit. The cause of this lowering of temperature is easily imdei Pieces of ice have generally some water adhering to them, and if salt be added to this ice, it is dissolved by the water and more ice melts. The necessary heat for this process, viz., the neat of solution and the latent heat required to melt the ice, n extra. from the mixture itself whose temperature consequently falls own. 126 PRODUCTION OF LOW TEMPERATURES [ CHAP. This is the principle of pvezing mixtures. This process, ho owever. cannot go on indefinitely. Fig. 1 shows the freezing curves "ob- tained with ammonium chloride, the ordinate representing tem- perature in °G and abscissa V concentration of the salt. When the salt is added to ice the tem- perature of the mixture chan as represented by the Hine AB 1 the eutectic temperature of ~]5<8°C is reached. Tempera- tires lower than this cannot he obtained in this way for when more salt is added it no long into solution. The curve AB represents equilibrium bet- I'tff. L— iN-tua aad water ■ ■ .-, ; e n solution and ice while CB ™n i i .. ^ „ , represents equilibrium between salt and solution and B denotes e eutectic mixture with a Table 1. — Freezing Mixtures. ed composition and fixed tem- perature. In Table 1 the composition of the eutectic mixture and the corresponding eutectic or c- tempera wnmoner salts, i rally hydrated salts are employed ! in that case the oorre! ing quantity hydrated salt Id be obtained by calculation, i raperatures repre- the lowest temperature that is possible to attain with that Freezing mixture, (if) Boiuwg 4 Liquid Under Reduce Pressure 4. Lot ntme may also he attained by allowing a lie boil under- reduced pressure. When a liquid evaporates it requires U for conversion from the liquid to the gaseous state (latent heat of vaporisation) Thus one gram of water at 100*C requh calories for complete evaporation. If such liquid be forced by so] conmvancc to evaporate rapidly and if the liquid be isolated." rh." ig may be produced. The oldest contrivance for utilising this process is the crvo- Wicated in Kg. 2. The bulb B contains water or some volattle liquid and the rest of the space is filled with the lap u? 1 Aulr. En:. Salt ! saJtperlOO tempera- , grams of the lure. rture. ; o 4 19-7 -^f ZnS0 4 27-^ .6-5 KC1 19-7 - T T • 1 NH 4 CI 1 5 - S iNO s 41-2 -17-4 o 8 37 -18-5 NaCl - 21 -2 : :i a 21*6 -33-6 CaCL 29 -B -- V. KOH 81*5 -<o5 VAPOUR COMPRESSION MACHINE 127 of the same liquid. When A is immersed in ice the vapour in it con- denses and the pressure of the vapour in B becomes so much lowered that the liquid 'In B boils. The latent heat necessary for this purpose is ex- tracted from the rest of the liquid which consequently free. Xow-a-days this principle is em- ployed in a huge number of refrigerat- ing machines both for industrial and domestic work. Water, however, is not a suitable liquid to use for though it has a large latent heat of evaporation, the vapour pressure at low temperatures Fig. 2.— The Cryophorus, is small The liquids commonly employed are ammonia, sulphur dioxide, etc. Two types of such machines arc in use : — (1) Vapour compression machines, (2) Vapour absorption machines. The vapour compression machines are more efficient, particularly for large plants, and require less initial cost; consequently, their use is 'more common than that of the other. The only essential difference between these two types of machines consists in the manner of compressing the low pressure vapour. In the former a motor compressor is used while in the latter a dilute aqueous solution at ordinary temperatures is employed to dissolve the low pressure vapour and the coneentral solution is heated in a generator to expel the gas at high pressure. We shall now describe these machines in greater detail. 5. Vapour Compression Machine.— Fig. 3 shows the essential parts oE a vapour compression machine. There are three principal parts Water ou P=1L Etmos js , CoMsto apftse p=2'5 atmoa. p~j] Liquid ammonia pffil High pressure ammonia ,ow tiresaure ammonia V Wig. 3.- Essential parts of a Vapour Compression Macbin the compressor P, the condenser C and the refrigerator or evapo- rator R. The cylinder of the compressor has two valves, S and D, .28 PRODUCTION OF LOW TEMPERATURES [ CHAP, rhe former for the suction of the low pressure vapour from the eva- porator and the latter lor the discharge o£ the compressed vapour to th<- condenser. When the piston p moves upwards the pressure in the cylinder falls below the pressure in the evaporator and the l™ pressure vapour Is sucked in through S and the suction pipe. " Dicing the downward stroke Lire vapour is compressed and then delivered to the condenser C through the discharge vake D and the discharge pipe. The condenser is cooled by cold dilating in the outer chamber. On account oE the low ^temperature and high pressure the vapour liquefies in C. This liquid passes through the expansion valve or the regulating valve V which is simply a throttling valve to reduce the pressure of the liquid from the high pressure prevailing in the condenser to the low pressure m the eva- porator Due to die low pressure the liquid boils thereby extracting its latent heat from the cold storage space surrounding the eva- porator. This space is consequently cooled. In some cases the evaporator is surrounded by brine water kept, in circulation. The brine water thus becomes cooled and is taken elsewhere for reFrige- rati i purposes. The low pressure vapour is sucked in by the com- pressor and the cycle or operations continues. In the diagram anhydrous ammonia is supposed to be uti efrigerant. The pressures and temperatures of ammonia In of die apparatus are approximately as indicated in the . i : on machine is shown in Fig. 4 w hicl Iphur di 13 the refrigerant. Vapour ively employed in ice-making. t and other foodstuffs and for various other Indus- trial purp 6. Refrigerants.— Various liquids have been used as refrigerants, important one ai imoniaj sulphur dioxide, ethyl chloride methyl chloride. Of these ammonia is most commonly used in refrigerating plants, while sulphur dioxide is employed in many household , There are various criteria for .selecting a suitable refrigerant : (1) The latent heat of the refrigerant should be large so that the minimum amount of liquid may produce the desired refrigerating effect. (2) The refrigerant must be a vapour at ordinary temperatures and pressures but should be easily liquefied when com-. d and cooled. Generally a temperature of about 5°F (some- below ice-point) is required in the evaporator coils and about (about room temperature) in the condenser coils. (3) The pres- 1 of the vapour of the refrigerant in the evaporator coil must be greater than the atmospheric pressure, so that atmos >' impurities may not be sucked inside and later block the valves. With cooling at room temperature surrounding the condenser the pressure necessary to liquefy the gas in the condenser should not be too large otherwise the compressor and the cylinder will have to he made very- stout and consequently costly and there will be much leak g vapour into the atmosphere/ (4) The specific volume of the vapour I VI.] REPlUOERANTS 129 of the i ••:: 1 l should not be large otherwise a very large compressor will be necessary, . 7 h - :: important pii | me common refrigerants are given m rable 2. V, is table it k evident that ammonia is the gerant. One pound of ammonia will produce the same amount of refrigeration as 8.75 pounds of carbon dioxide while the pressure hi the condenser in case of carbon dioxide is about ater than in the case of ammonia. In the matter of spin j! 1 volume, however* carbon dioxide possesses an advantage. Sulphur dioxide requires a less stout compressor and condenser than rua but for the same refrigerating effect the compressor ha be made large. Table 2. — Characteristics of refrigerants* Ammonia Sulphur Carbntl ' dioxide Methyl irjde Freoti CCJiF. 1. Boiling point in °F at a tm. pres- sure -28.0 14.0 -108.4 -10.6 -21.7 2. Latent heat of 1 \ aporation at 5 in B.t.u. per pound 565,0 175.fi 114.7 178.5 69.5 5. Refrigerating effect in B.t.u per lb. 474, 141.37 JUS/;! 148.7 5 LOT -L Vapour pressure at 5°F in lb/in s . 34,27 11.81 33-1.4 20.89 26.51 5. Vapour pi esi at 86°F in lb/in* 6. Specific volume of vapour in e vapor? cm ft. per lb. 7. Horse-power for a ifrigei Ei n ] 200 B.t.u. per min. 60,45 10B9.0 6.421 0,09 >3.2 95.53 4.529 1.06 107.9 ].-1S . : : * A number ants have been 180 PRODUCTION OF LOW TEMPERATURES [CHAP. This method mav be employed to obtain extremely low tem- peratures by using liquid hydrogen and helium. These liquids are Sowed to boil under reduced pressure when temperature* lower than their normal boiling points are reached, 7. Electric Refrigerator,— Fig. 4 shows a modern electric refri- . l . Y? - ' „t „:«..-■ .nwlin™ iiitniYMi-irailv on r. he vaUOUr (Oil Liatricl SO^ I High pressure S0 3 I Low pressure S0 3 Fig. 4-— Frigidaire, The refrigerator coil R contains liquid sulphur dioxide which extracts he:--!, from the surrounding space and evaporates and the low pressure vapour collects at the top. This vapour communicates i wtti i the suction pipe S and the crank case K to the motor switch W. When eno has collected in die top of R it exerts a large pressure -which is transmitted through S and K thereby operating the switch VI. AMMONIA ABSORPTION MACHINE 131 '.,'.'. This starts die motor and the latter works the compressor P, as a result of which, the low pressure vapour is sucked in through S to the ciunk cs and compressed by the piston and delivered to the condense] C. The condenser is cooled by a current, of air forced across il b) the fan mounted on the motor M but in some cases the cooling is bi ought about by the flywheel itself whose spokes are shaped like the blades of a fan. The high pressure sulphur dioxide vapour on his cooled liquefies and collects in the reservoir T. From here on > l of the high pressure the liquid is Forced up through the liquid pipe L and enters R through the needle-valve N. When enough liquid, has collected in R the float valve V rises and closes the valve N. Thus the machine works only when the gas essure in R becomes large and liquid is transferred from the storage ik T to R only when the quantity of liquid in R becomes less and float valve has sunk so as to open the needle valve N. 8. Ammonia Absorption Machine. — 4j> already slated the ab- sorption machines differ from the vapour compression ma- chines only in the manner of converting the low pres- vapour into high pres- sure vapour. Ammonia is the most suitable refrigerant use in absorption ma- chines, and water is a very suitable absorber. Water at F absorbs about one thou- i id times its volume of ammonia vapour but when the aqua ammonia 'solution is freely escapes from the solution. The wolfing of >an ammonia absorption machine will be easily understood from Fig.J5. The generator A contains a strong solution > nonia gas in water and Is heated by a burner as shown in the or by means of pipes carrying steam. Ammonia gas is expelled from the solution and passes into the coils immersed in the con- denser 1J through which cold water continuouslv Hows. The; gas idenscs there under its own pressure into liquid ammonia. The liquid ammonia thus formed passes through a narrow regulating valve to the spiral immersed in the refrigerator C, where on account oE ili low | i me it evaporates. The valve is adjusted to maintain desired difference of pressure on the two sicles. Through the refrigerator flows a stream of brine water which becomes cooled by the evaporation of ammonia. The cool brine colution may be taken to .il i \ place for refrigerating purposes. mmonia gas formed in- the coils in C is absorbed by water i diluta ammonia solution contained in the absorber D arid thus the pressure is kept low. The solution in D hecomes concentrated and is transferred by the pump P to the generator at the top. Thus I" ... 5.-— Ammonia Absorption Machine. heated to 80° F ammonia vapour 132 PRODUCTION OF LOW TEMPSEEATUREfi P the supply of concentrated ammonia solution is kept up. Dilute ammonia from tin bottom of the generator may be run to the ab sorber and concentrated. Thus tlie cycle is repeated and the action is quite continuous. The difficulties of this machine are that It h a m\\ efficiency and the pressures are widely different in the condenser and the orator. The low efficiency h due to the circumstance that the heat absorbed by the ammonia in the generator is much larger than the heat absorbed by it in the evaporator coils. Further the machine has a moving part in the pump needed to transfer die liquid to the generator, and is costly. All these difficulties are avoided in a clever invention by two Swedish engineers, von Platen and Muuters, which is :<3 in the market under the name Elec- trolux Refrigerator, hi this Dalton's law of partial pi. is used to make the tola tire in the condenser and the evaporator equal, maintaining at the same time a difference in partial pressures of ammonia in the two chambers ; this is accomplished by using an inert gas like hydrogen at a pressure of 9 atmos., the partial pres- sure of ammonia being •:! atmos. in die evaporator and the absoi and ammonia liquefies in the condenser at the pressure of 12 atmos. Concentrated ammonia solution is forced up into the generator by heat and not by a pump. (iii\ ^diasatic Expansion osf Compressed Gases 9. Cooling produced by the sudden adiabatic Expansion of com- pressed gases. — ■ Idenly allowed to considerably on account of the k it docs in expanding, vide' § 23, Chap. II v ////?) 'v- ' ■>'>: The cooling: may he so great that the ga.s may eve [£y. An example which is easily available in a big town is afforded i irbon dioxide. If such a cylinder be sud- denly opened and a piece of cloth held before it, the issuing ■ deposited in the form of solid CO a (called dry ice commercially) . This principle was utilised by Cailletetwho first liquefied oxv in 1877, He compressed oxygen to a pressure or 300 atmosphere j capillary tube cooled to -2SPC by liquid sulphur dioxide boil rider reduced pressure and then suddenly released the pressure. A mist, of liquid oxygen was formed which disappeared in a few seconds. PicteL compressed oxygen to a pressure of 500 atm. and ec it to about — 130°C by liquid carbon dioxide evaporating under re- re. Then he suddenly released the pressure. Oxygen En the form of a white solid was thereby obtained. Tn 1884 Wrob obtained a mist of hydrogen while in 1S93 Olszewski obtained liquid tin ient quantities, by cooling 'compressed hydrogen with liquid oxygen and then suddenly releasing the pressure. Simon in 1933 produced appj quantities of liquid helium by sudden adia- •jiiic expansion of the compressed gas which had been precooled by solid hydrogen evaporating at reduced pressure. VL] [PACTION OF GASES 133 The process is, however, essentially discontinuous, hence for commercial purposes it was almost discarded; but a novel wr. utilizing the principle has been invented by Claude and Heylandt foi I air (see sec. 22) » W; '"escribed above two types of refrigerating machines. There is a third type also which is sometimes employed. This may be called the air compression machine because air is here used as the [rigerant. In dlis air is first comprcsssed in a compressor, the heat of co ion is then removed by passing the gas through coils kept , cold circulating water. This cool compressed air then suffers adiabatic expansion in the expansion cylinder and becomes consider- ably cooled. This cold air then traverses the cold storage space and thu Mated and is again compressed. Thus the cycle continues. This is the principle of the Bell-Coleman refrigerator largely used for the refrigeration of cold storage chambers in ships. (h) COOLING DUE TO PELTIER EFFECT 10. It is well known that when an electric current (lows in a circuit from bismuth to antimony through a junction, this junction is ooled. Ellis is known as the Peltier effect and may be utilized in tducing cooling. This cooling is rather small though semi-conductor thermo-junctions have recently produced much more cooling, and have been employed in some refrigerators. (*■•) Cooling by Joule-Thomson Expansion 11. This method, is of considerable importance and will be con- sidered in detail later in this chapter. (vi) Cooling Due to Desorftion I2» Charcoal adsorbs a number of gases which are released on lowering the pressure, and when these gases escape, a cooling results a manner somewhat analogous to the case of evaporation of liquids. is is called the "Desorption method" and was utilised by i in. In an experiment charcoal adsorbed helium gas at 5 atm. I0°K and subsequent de sorption to 0.1 mm. pressure lowered the temperature to 4 P K which is sufficient to liquefy heliu: l. LIQUEFACTION OF GASES 13. Liquefaction by application of pressure and low temperature. — which are gaseo^ at ordinary temperatures may be i the liquid state if'they are sufficiently cooled, and sirnul- - pressure be applied to the mass. When pressure the molecules come closer together and if heat motion sufficiently small, they may coalesce and form a liquid mass, ll the liquid so cooled be allowed further to evaporate rapidly, Lem| nines may be obtained. The production, of in' I leratures is thus intimately connected with the . ■ i : hi of such gases which ordinarily show themselves efracl 134 PRODUCTION OF LOW TEMPERATURES [CHAP. compression was The earliest scientist to try this effect of combined cooling and Faradav* who, as early as 1823, employed the apparatus shown in Fig. o for liquefying chlorn One end of the bent glass tube contains dr.: substance from which chlorine is liberated by heat while the other end is immersed in a freezing mixture. Gaseous chlorine coltecLs in the other end and finally lique- fies under its own pressure. By applying this process Faraday and others suc- ceeded in liquefying a large number of gases but some viz., oxygen, nitrogen, hy- drogen, carbon monoxide and methane baffled al! at- tempts at liquefaction, to S000 arm. were used. Fig. &— Faraday* :itus iot Liqpefeclton of Chlorine, carro in 1!- abject And i' w abov to which it is .subjected. though sometimes enormous pressures up They were, therefore, termed menl gases. Discovert of the Critical Point— Ths cause of these failures "be- ta Andrews' discovery of the critical temperature has already been treated in Chapter IV. arty showed the importance of the Ti, ■ 1 1 clea rly that, for ev er y subs tance , Wchmudlyo'ccurs in the gaseous form, there exists a temperature bove which 'it cannot be liqueSed however high mas' be the pressure , w h subjected. Hence, in order to liquefy a gas by this method it must be pre-cooled below its critical temperature. The determination of the critical point (p, 100) is not, how- ever, easy. The early workers did not in fact, wait for its deter- mination. They cooled the gases by ordinary methods as much as they could, add then applied high pressures. ' 14. The principal methods of liquefying -air and other gases are the following:— (1) Pictets cascade method which utilises a series of liquids willi successively lower boiling points but the principle is the same as explained above ; (2) the Limle and Haropsons methods employing the Joule-Thomson effect; (3) the Claude and Heylandt methods which utilise the cooling produced when a gas expands doing ♦Michael Faraday (1791-1867), the "prince of experimenters, was bora of htunbto i - to London. At the age of thirteen he became errand boy to a •.-Uer but later in 1813 he got employment under Sir Humphrey Davy at the Royal Institution where be carried on his scientific work and finally succeeded Davy in 1827 as Director of the Royal Institution. His greatest work ls the dis- covery of electromagnetic induction in 1831. VI.] SERIES KIlFRIGERATION 135 external work. We shall now consider these methods in greater detail. 01 these the first is historically the oldest and theoretsqjdlv the most efficient hut is somewhat cumbersome and is very little used at present. 15* The Principle of Cascades or Series Refrigeration. — The method was first employed by R. Pictet in 1878. In principle it may be described as a number of compression machines in series. Pictel employed machines containing sulphur dioxide and carbon dioxide and obtained temporarily a jet of liquid oxygen by allowing com- pressed oxygen to expand adiabatically. Wroblewski and Olszewski at Cracow obtained sufficient quantities of liquid oxygen, nitrogen and carbon monoxide by the cascade method and determined their properties. Olszewski used ethylene as another intermediary below carbon dioxide and could thereby cool these gases below their critical temperature. Kamerlingh Onnes later employed die combination of methyl chloride and ethylene for liquefying oxygen. The prin- ciple of the method is illustrated in Fig. 1. Machine (1) utilizes methyl chloride. This has got a critical temperature of M3 C C and hence at room temperature it can be easily liquefied by the pressure of a few atmospheres only. Water at room temperature flows in the liquid [lea, a Fig. 7. — Illustration of the Method of Cascades. ket in (1). Liquid CH 3 C1 falls irrto the jacket in (2) which is connected to the suction side of the compression pump. Thus the Liquid evaporates under reduced pressure and its temperature falls to • hi 90 Q C. The compressor returns compressed CH 3 CI gas through the tube inside jacket (1), which is shown straight but is really in the form of spirals. Inside the jacket (2) is placed the condenser coil through which ethylene passes from the compressor or the gas cylinder. It liquefies and i!i. . enters the jacket (3). There it evaporates under reduced PRODUCTION OK LOW TliMPJERATURES [CKAP. ire and lowers Che temperaAiwre to about - I60°C. Tbj tube insi acket (3) oxygen from a and liq -essure. Liqu ollected In Ehe sfe D. The lowest temp;. obtained by boilin en under redt pressure is -218°C which is higher than tl matures of (-228.7), hydrogen (-240*0) and helium, hence the method of cascades failed to liquefy these three gases. The method of cascades is very useful for laboratory purpc The use of the compressor can be entirely dispensed with by the use uable liquids boiling under atmospheric or red essure. : suitable liquids can be selected from Table . res the normal boiling point, the critical temperature and the tri int Cor the common gases. The interval between die critical temperature and the triple poirj icnts the range in which the liquid is avail- able. Still lower temperatures ran be obtained by further reducing the press. ■ the liquid when it solidifies. Thus, using solid nitrogen and a good pump a temperature of about -2 an be obtaim :] -above the critical temperature qf neon and it is not possible to bridge the gaps a nitrogen and neon and between hydrogen and' helium in this way. ■ Table 2— G '", normal boiling point and tri ases Sul B. P, at 1 atm. Critical tem- Triple point pres s per:: . Mechvl chloride - 24.09°C 143.3 - 10S.9 C C Sulphur dioxide - 10.1 157 lOnia 131.9 -77.7 Car; :ide -78.6 31.0 -56.6 Nitric oxide 89,8 36,50 - 102,3 Ethylene -in- . 9,50 -169 ane -161.37 2,85 -183.15 Carbon monoxide . -.190-0 -13 Oxygen - 182.95 _m -218.4 Nitrogen -19! -W -209,86 . -245 -228.71 -2' Hydro -252 -239.9? -259.14 Helium -267 16. Production of Low Temperature* by utilizing the Jonle-Thomcon Effect. — As the above method is not capable of liquefying hydrogen and helium, another process began to be utilized from 1898, This Is the Joule-Thorason effect discovered in 1852. A fall mathem VI. THEORY OF POROUS riAIG EXPERIMENT 187 analysis of the phenomenon is postponed to Chapter X. We describe the phfnomem n hare: We have already described Joule's eriment (p. which showed, that for permaneja the internal em tot depend upon volume, "I.e., t-jp | = 0. This is called Joule's law or Mayer's hypothesis and is the characteristic property of a perfect gas (for perfect monatomic gases U — ^NkT per rnoIA But Lids is not strictly true [or idle actual gases of nature ; they all show deviations from the st: te of perfectness and hence for them U is not independent of e. A slight change in the temperature of the gas should occur in joule's experiment, but since the capacity for heat of the contained air" is negligible as compared to the heat capacity of the surrounding , no change in temperature could be observed. In 1852 Lord Kelvin, in collaboration with Joule, devised a modification or Joule's experiment in which very small changes in temperature produced by expansion could be easily measured. This is called the "Porous plus experiment and provi an unfailing test of Mayer's hypoi Widi its help we can easily find how far a gas deviates from the state of being perfect. Before proceeding to describe this experiment we shall discuss the theory underlying it. 17. Elementary Theory of the Porous Plug Experiment.-— In this experiment a highly compressed gas is being continuously forced at a constant pressure through a constricted nozzle or porous plug. The as its name implies", consists simply of a porous material, say, cotton-wool, silk, etc. having a number of fine holes or pores u thus equivalent to a number of narrow orifices in parallel. TT during its passage through tlie pores becomes throttled or wire-drawn, viz., molecules of the gas are drawn further apart from one another doing internal work against molecular attractions. This is always the case whenever a fluid has to escape through a partly obstructed passage. On either side of the plug constant pressures are maintained, the pressure on the side from which the gas flows being much greater than on the other side since the plug offers great resistance to the flow of the gas. This expansion is of a character essentially different from Joule's e ision. In Joule expansion, the gas expanded with- out doing any external work. Here it expands against a constant externa! pressure and hence it has to do external work also, together with any internal work, while some work is done on the gas as well, plug is surrounded by a non-conducting jacket so that the process is adiabatic in the sense that no ""heat enters or leaves the system. For such 'processes we now proceed to show that the total heat function h = u -{- pv remains constant. To prove this theorem, let us consider a. mass of the fluid ous plug C from left to right as indicated in Fi .. ,-.. 7" A , u A and p E} v Bf T B , u a be the pressure, volume, temperature and internal energy of one gram of the fluid before and after traversing the en":; re: respectively. Suppose that one gram of 138 PRODUCTION OF LOW TEMPERATURES I CHAP. L Vh C -•/.: M the gas is contained between the porous plug and some point. M on the left and also between the plug unci some point N on the right- For visualising the process we may assume an imaginary piston A at M separating this quantity of the gas, and that the flow of the gas is caused by the {or ward motion of the piston A. Actually, however, the rest of the gas exerts a pressure p± at M which is maintained by the source of supply. The gas after travers- ing the plug pushes forward the imaginary piston B whose motion is opposed by the pressure p of the gas to the right of B. The initial and final states are she respectively at (a) and (b) in the figure, the initial volume of the Fig. &,— The Porous Plug Experiment. .gas being equal to ML and the final volume equal to ON, The gas during its passage through the orifices in the plug has to overcome friction, viscosity, etc., and hence loses energy. The escaping gas issues in the form of eddies and its temperature falls cor ily just at the jet (and this effect is spurious) because some th'. i rgy is now converted to tile energy of mass motion. The x, subside after traversing a short distance and the .i.ture consequently rises. Let us consider only the steady and after transmission through the orifice, i.*?.., at points he plug where eddies are not present. We assume lowly so that the energy of mass motion is very sural! and negligible in comparison t nergy of thermal motions. Now our initial system is ML and the final system is ON. But the plug OL is initially and finally in the same state ; hence the change simply consists in a change from MO to LN* If in addition, the tube is surrounded by a non-conducting material, no heat is supplied to or withdrawn from the system. Some work is, however, done by the external forces on the slowly moving system. The force acting at A is equal to p A )< S where S is the cross-sectional area of the cylinder. The work done by this force upon the i: ' ;' } a)< 5 X MO = p x v A . Similarly, the work done by the gas in forcing the piston B is p^v*. Therefore, the net work done by the gas is put's — PaV*> and from the first law of thermodynamics, since A(> — 0, tile work done by the system is equal to the decrease in its internal energy, viz., »a -"b=Mb^Ma (i) OT *A +J*A»A = »S+J&B»B .. (2) Hence u -f- pv remains constant in the throttling process. "0 THE POROL'S PLUG EXPERIMENT 139 For perfect gases Boyle's law flw) T ^ constant and Joules law u^=cT hold true, and therefore u-\-pv would depend upon tempera- cure only. We have just shown that u+pv for the porous plug experi- ment is constant whether the gas is perfect or not. Hence, tor perfect oases the temperature on both sides of the plug would remain the same. In actual experiments, however, a cooling- effect was observed £or most gases such as air, 2 , N 2 , CO, and a heating effect Thus none of the gases examined was peiletf. inc. be due either to deviations from Boyle s both* If we know the former, (p. 97) we am find the latter experiment. If at the particular pressure and temperature 1 1 A, the gas is more compressi than <*);. alone ;as is Upon from work in case of Hj. lack of per redness may law or from Joule's law or as from Amagat's experiments by performing the porous^ plug- pressure and temperature in A. at lower pressures [c£, curves of N* C0 2 betore the bend ( & z?* <PuVn. Therefore, due to deviations from Boyle s k ut <ii i-e. the gas would show a cooling effect*. IT the l?ss compressible [cf., H a ) there would be a heating effect, these effects will be superposed, the effect due to deviation Toule's law. Since in actual gases, cohesive forces are present will be done in drawing the molecules further apart during expansion and the gas would become cooled. Thus the jouleThomson effect due to this cause would always be a cooling effect. Ihe observed effect is the resultant of these' two effects and may be a heating or cooling effect depending upon the sign and magnitude of the former effect/ 18. The Porous Plug Experiment,— We shall now describe actual experiment. Joulef and Thomson were the first to carry out these experiments. They employed a cylindrical plug, which is indicated in' Fig, 9. The compressed _ gas Hows through a copper spiral immersed in a thermostat and after having acquired its temperature, passes through the porous plug W* The plug consists of silk, or cotton-wool or other porous material, kept in position between two pieces of wire-gauze and enclosed in a cylinder of some non-conducting wood bb. The plug and part of the tube is sur- rounded by asbestos contained in a tin cylin- der zz so that no heat reaches it from the bath. Joule and Thomson worked with air, 2 , N 2 . CO,, between 4° and 100°C, the initial and final pressures being 4.5 atmospheres and 1 atmosphere respectively. Joale-Tnom- soh's porous pirns; of two energy '♦It is important to remember that the internal energy is made up parts ' ) kinetic energy which depends upon the temperature, (..•- 1 due to molecular attractions, the former being much greater in a gas uik ordinary conditions so that Joule's law is approximately obeyed. { foalej Scientific Papers, Vol. II, p. 216. 140 PRODUCTION OF 1X.AV TEMPtiRAi CHAP. Some of the subsequent worl^ers employed a plug of the "axial flow" type as used by Joule and Thomson, while some others err, i plug of the "radial How" type. In the latter the gas flowed from the outer side of a hollow cylindrical plug to the interior and hence heat insulation was better. Certain others employed only a throttle valve or a restricted orifice. From these experiments Joule and Thomson found that the fall in temperature was proportional to the difference in pressure on the two sides of the plug, i.e., bd—k(}> A —f! u ), where k is a constant, characteristic of the fluid. They found empirically that k — A/T* where T h the absolute temperature of the gas, while Ross-Innes found k = A -I- B/T, Hoxton, however, found that his results were represented by the form. £_.4+-j'~+^j (*} :]-?-0 We can find a value of k from theoretical considerations i Chap. X) , If the gas be supposed to obey van der Walls' equation of state, it. can be shown diat the Joule-Thomson ef r Exercise .[.— U-; dues of a = 1-36 X MP atm - cm * an d & = 32-0 c.r. for a gram-molecule gen at N.T.P. and C* = 703 cal. akulate the Joule-Thomson effect from equation (i) , 9 = I_ -0 1 x 10* A Ap 7.03x4.18xl0 7 L 8< ^73 ; 4. 18x10' d 01 X 10* _dcg. 7*03x4.18x10* atrm/cms" 1 r a van der Waals' gas the cooling duced in the Joule-Thomson process. I 11 * •' by the gas— 1 * ^dV = — — — , , • . Net work (external -[- internal) done by the gas pro- " =^f7"i-r 1 )+A( J fr i - A ) + ::■■■ 2a since from van der Waals' equation /?F — RT 4-bp — (a/V) approx lor the two sides of the plug. will produce a cooling by _ &q (since AC? - 0) such that (see Chap. X, sec. 10) - VL] PRINCIPLE OK KPGENK.RATIVK COOLING 14? Wb'jt f ^fe or -C B AG - #A0+*AH-£j= (—Art a PP Combining this i b on C„ - C B t= R, we get equation (4) . 19, Principle of Regenerative CooUns-— The Joule-Thomson « in s observed tor most gases is v. as, £<w a jr at a tem " oerature 20°C when the p iun on the two s.des are SU atmospheres and 1 atmosphere respectively, Joule and Thomson found that" ike tei;. re falls by Il-VC. Hence the method eduid not be employed for a long time tor producing liquefaction. Subsequent however, it was discovered that the cooling effect can be unengi! by employing what is called the reg ■ A portion ol tne eas which first suffers Joule-Thomson expansion and becomes mp] wed to cool other portions of the incoming gas betore die la- bes the nozzle. The incoming gas becomes still m * cooled after traversing the nozzle. In this way the cooling eltect can be made cumulative. In actual practice* this is secured by using either concentric tubes as in Linde's process or by means of Hampson spii r : i . Two or more concentric tubes are arranged in tlie form of spirals, die inner one carrying the high-pressure inflow gas while die outer one die low-pressure outflowing gas. In the regenerative method a further advantage is gained by the fact that the lower the ture the greater is the Joule- Thomson cooling. The regenerative principle is illustra- ted diagram matically in Fig. 10 where the high- pressure gas from the compi enters the spirals contained in the water- cooled jacket A. The gas next enters the regenerator coils at E and by expansion at the valve C becomes cooled by a small amount. This returns by ,thc outer tube abstracting heat 'from the high pressure gas, and reaches F almost at the temperature as the incoming gas at E. The As is again compressed and cooled by A and re-enters at E. As time passes, the gas approaching C becomes cooled more and more till the Joule-Thomson cooling at C is sufficient to liquefy it viz., its temperature reaches the value at which the gas would liquefy under the pressure prevailing at F. A portion of the esc::: i j gas then condenses inside the Dewar flask D. At this stage the temperature throughout the apparatus becomes steady i . ic represented by the curve shown in Fig-, 11, p. 142. The part L M" represents the continuous decrease of temperature of the as we approach the nozzle through the inner tube while MN repre- sents die Joule-Thomson cooling. NL represents the temperature of the low-pressure gas which is less than that of the adjacent high- f -:i; i ;l Fig. 10. — Illustration of the Regenerative Coding. 142 i'JiODUCTION OF LOW TEMPERATURES CHAP, pressure gas. Thus the cooler low-pressure gas abstracts heat from die incoming stream. 20. Linde's Machine for liquefying Air,— The principle of regeneration (applied to heat- ing) was discovered by W. Sie- mens in 1*57, hut its application to cooling' processes came later. Linde* in Germany and Hamp- son in England almost simulta- neously (1895) built air lique- fying machines based on the above priii rip J e. At the present time, such machines have be- come quite common and many ies in tire world arc fitted up with such machines. Fig, 12 3 a commercial form of Linde's machine; e, d, is a two, or better ]}Js!anc« along (he Mgciicmlve tail 1ml\ II, — Temperature distribution 1 ... 12.~Linde's apparatus for liquefying air, three stage compressor, the machine e compressing the gas from 1 to 20 atmospheres while d compresses it from 20 to 200 atm, A charge oJ atmospheric air is taken in at. e and compressed by e to 20 atm. * K:ir ' y» Linde, born in 1842, was Professor at Munich. He published an account of bis air- 1 . machine in 1895. VI.] AIR LIQtttiFIERS 143 then cooled by passage through water-cooled tubes and is ^ to the suction side of the second stage compressor. The tomprebsed L d!en pa^es tough the cylinder f which contains caustic soda. J his a^o?b tie carbSn dioxide (IE this is not done, carbon dioxide ifflbSS? Nidified and choke the valves in the ^^Hggg; ■ -as then pusses to the tubes g which are cooled by a freeing m ixttne to . -20'C, From here it, passes on through the metallic tubw p^hiner coils of the liquefier proper. At a we have got the plug <in the first stage to 20 aim.), tb* temperature falls to about -<B <£ ind the air agali passes through the outer cods cooling the incoming t»- it is then led Sirowh the pipe P, to the second stage compression fnder where it is again compressed and allowed to pass through X reftigerator and thl inner coils to a. After the ^p etion o a few cycles the temperature of the incoming gas falls so low that the xerond throttle valve is opened The air .s now allowed to exp tndto 1 atmosphere when it becomes liquid and ^ ct %^j^^ .: from which it can be removed by the siphon A. The unliquefied gas is aeain led back through the outermost cods to the compressor e as nSd by the arrowt Fresh charge of air is beifg continuou ta^en in at*, compressed and delivered along with the gas from the middle tube to d. The process is cyclic. 21. Hampso^s Air Liquefier. This liquefier also lujlizes the joule-Thomson effect and the regenerative principle, but differs i from Linde's apparatus in details of construction. The special feature about it is the Hampton spiral. The high pressure inflowing ^as Passes through copper* tubes coiled in the form of concentric spirals arranged in layers; it then suffers the throttle expansion and becomes cooled This cooled air vises through the interstices between the a era or spiral and thereby cools the incoming li^h-pr^iire gas. After some time the high-pressure gas becomes sufficiently cooled so that on suffering the throttle expansion, it liquefies. The apparatus thus differs from that of Lmde only in the manner of cooling the m- comine 2as. The apparatus was later improved bv Olszewski. 1 he ITampson construction has been utilized later by Dewar, Onnes and Meissner for liquefying hydrogen (see Fig. 14) and helium. 22. Claud* Air Liquefier*- Although the Linde and Hamp: lynchers just described are in extensive use in laboratories and com- mercial installations, the machines cannot be said to be satislactoi ainly because the efficiency of the machine, t.e. f beat extracted/ wiergy consumed bv the machine, is extremely low (about 15%). llie coolinir process employing Joule-Thomson expansion is really very tent. A more efficient machine could be devised if the com- ,. K as was made to expand adiabatically doing external work and thereby suffered cooling, The technical difficulties in construct- ,i apparatus for continuous liquefaction of gases by adiabatic expansion were overcome by Claude. The main difficulty consists in finding a suitable lubricant for the moving parts of the expan- sion cvlinder since the ordinary lubricants become solidified at these m en 144 PRODUCTION OF LOW TEMPERATURES [CHAP. low temperatures. C d) Low Pressure Air Fig. 13. -Claude-Heylaiidt s;. laude utilized petroleum ether as the lubricant. This ... i . iscous at tempera- Canvm* tures of - 140*C, or even - 160*C and thus acts as an effective lubri- cant up to this range. 13 shows diag i i the Claude's air liquefying: chine, The gas (ram the com 5 divided Into two parts to the expau- cylinder and suffers adiabatie onsequent coo] In this expansion it does work which is utilized in doing work on the compressor. The :d gas traverses upwards in ihe pipe D thereby coi i he second n of the Incoming com : in the second heat-exchanger. The high-pressure gas thus parti- ally liquefies. It then suffers Joule- Thomson expansion at the throttle' valve. The evaporated r taken to the compressor and afi Theoretically ( I ulc * be more efficient than the only slightly so. This is ilties of • ide method, The 1 has the great advantage that no movable parts of the ad hence the csmtructia dt slight- Lqucfier. U ; taiuion turbine will po- vera! dprocariug engine, Kapitza in 1939 developed ier in which the compressed air is allowed to expand ire of abc ; ressure of about 1-5 atmos, and drive a turbine wheel, and thereby suffer cooling on account of ie. The machine is about three times as efficient as Lmde's li is it utilizes a pressure of about 5 atmospheres only, all danger due to high pressure is eliminated. 23. Liquefaction of Hydrogen.— The method of cascades failed* to liquefy hydrogen, Wrohlewski thereupon studied the isotherms i gen at low temperatures, ai that caL the values and b, and thence the critical constants (p. 95) . The critical 1. unci to be very low (— 240°C.) and Olszewski , ted out that this temperature could not be reached by the evapo- n of liquid nitrogen, which was the most intern shen known. The physicists then turned to the Joule-Thomson This !• however, first appeared to be inapplicable to Regnault had shown that when hydrogen ♦Neon can be utilised but it is very rare. VI. HYDROGEN LIQUEFYING APPARATUS H5 is subjected to this process, it gets heated instead of being cooled. That the difficulty is not fundamental and insuperable can be seen from the expression for the Joule-Thomson effect viz. } • . ■ ■ (4) Ap-CART } It is thus proportional to 2a/RT-b but in hydrogen and helium a is so small that at ordinary temperatures 2a/RT<,bj and the term on the right-hand side becomes negative. Hence, for a negative value of Aft *"■*» expansion of the gas, &Q is positive at ordinary tempera- tures, and the gas shows a heating effect. If T be sufficiently reduced, the right-hand term in (4) eventually becomes positive and the gas shows a cooling effect. There is just a temperature where 2a/R.t- b — Q, i\fi v where Joule-Thomson effect changes sign j this temperature is called the 'temperature of inversion' T-, which is thus equal to 2a/ bR. This relation gives, after substi- tuting the values of a and b for hydrogen, T,- to be - 73 1> C Olszewski experimentally observed the Joule-Thomson effect for hydrogen at various temperatures and found the temperature of inversion to be - 80.5 fl C. Similar is the case with all gases. There is a temperature of inversion for all of them which, however, depends upon the initial pressure- of the gas. Even the Joule-Thomson effect, depends very much upon the initial pressure. It is thus dear that the behaviour of hydrogen and helium is not anomalous ; they differ from other gases only in having a low temperature of inversion. Now since T t » the critical temperature, is equal to da/27hR (p, 95) and Ti = 2a/bR f it follows that T i= (27/4) T t . This rela- tion is found to hold true approximately. Hydrogen must, therefore, be cooled below ~80 D C, for liquefac- tion. But "for practical success it should be pre-cooled to the Boyle point T B (p, 88) which is defined as the point at which — ^-— — 0. Calculation with the help of van der Waals" equation shows that this temperature T B =a/bR r Hence r B = £7V We thus see that hydrogen should be pre-cooled to about 96°K {- 177'C). This tem- perature in easily attained if we immerse the hydrogen liquefying apparatus in a bath of liquid air. 24. Hydrogen Liquefying Apparatus. — Dewar first succeeded in liquefying hydrogen in tliis manner in the year 1898, Travers later i 1 1 1 1 i : /red the apparatus. Hydrogen prepared from zinc and sulphuric acid is compressed to about 150 atrn, and then passed through coils immersed in water in order to deprive the gas of the heat Of compression. Next it passes through cylinders of caustic potash dehydrating agent and is deprived of its carbon dioxide and moisture. This is essential as these impurities would solidify much before the liquefaction of hydrogen sets in and choke the tubes. The 10 146 l'RQDUCTION OF LOW TEMPERA! L RES [CHAP, gas then enters the liquefier and traverses the regenerative coils A (Fig. 1-3) which are cooled by the outgoing- cold hydrogen gas, and in the final steady state, becomes cooled to about — I70°G. Next the gas passes through a refrigerat- . ing coil B immersed in liquid air, and then through another refrigerat : n .; coil C immersed in liquid air boil ing at a pressure of 100 mm. This is adjusted by allowing liquid air from F to trickle into G and by eva- cuating G through a pump attached ai r. The tempera tine of the iiv drogen gas thus falls to about -200°C. Alter this it traverses the coil D and suffers Joule-Thomson expansion at the valve a which is operated by H. The gas thus be- comes cooled and this cold gas- passes up round the chambers G and F, thereby cooling the coils D and C, to the chamber R and from there to the compressor. Thus ■ alter a few cycles the temperature of the incoming gas at a falls to -25Q D C, and then on suffering- the Joule-Thomson expansion it liquefies and drops as liquid into the Dewar ! I V. Fig. 14. — Hydrogen liqw Many later investigators devised apparatus which have a large output. Amongst them may be mentioned Nernst, Kamerlingh Onncs and Meissner. Dimes' apparatus h very similar to his helium liquefier. Meissner's apparatus is somewhat different in construction but similar in principle. Liquid hydrogen boils at -252.78 C C, under atmospheric pressure. By causing it to boil under reduced pressure it can be frozen to a white solid. 25. Liquefaction of Helium. — Helium could not be liquefied for a long time. The attempts of Dewar and Olszewski to liquefy helium by the adiabatic expansion method were unsuccessful Kamerlingh Onnes,* however, proceeded in his efforts very systemati- cally. He studied the isotherms of helium down to liquid hydrogen temperatures (up to— 250°C) and obtained the critical constants for helium. He found the following values :—T { = 5.25'K, p ( = 2.26 aim. and normal boiling point — "4.2f>°K. The joule-Thomson jnver- *He£ke Kamerlaiifih On ( 53— — 1926), bnrn in Holland, became Professor of "Physics at Leiden where he established his low temperature laboratory and investigated the properties, of substances at low temp vi.J SOLIDIFICATION OF HELIUM M7 sion point came out to be about •55°K. and the Boyle point 17 :: K. is temperature could, therefore, be readied by pre-eooling the gas with liquid hydrogen. Kamerlingh Onnes: was thus convinced of the possibility of being able to liquet)' helium by the Linde process. He u actually liquefying it in 1908 in his laboratory ac Leiden. Subsequently helium liquefiers were constructed at Leiden, Berlin and Toronto. To-day there are scores of helium liquefiers in the world. Since helium is rather costly, the arrangement should be such that it can work in cycles, In the apparatus used at the t pyogenic laboratory at Leiden gaseous helium compressed to 36 atm. is passed through spirals immersed in liquid hydrogen boiling under reduced pressure and then through outgoing cold helium vapour. The gas then suffers Joule-Thomson expansion and becomes liquefied. The plant for liquefying helium is, therefore, complicated by arrange- ments tor liquefying air and hydrogen. Both hydrogen and helium were liquefied by Kapitza in 1934- by the Claude-Heylandt method. Helium was liquefied by Simon by the adiabatic expansion method and also by the desorption method. Collins in 1917 developed a commercial type of helium liquefier based on the Kapitza method. In the Collins expansion engine the piston and cylinder are constructed of nitrided nitralioy steel, the clearance being about 0.0005 inches on the diameter and the operation being completely dry. Thus the leakage of gas is extremely small and whatever does leak, also goes to the suction side of the compressor. 26. Solidification of Helium. — Kamerlingh Onnes tried to solidify helium by boiling it under reduced pressure, but though he claimed to have reached 1.15 n K in 1910, helium still remained a fluid. In 1921 he again tackled the problem and by employing a battery of large diffusion pumps he reduced the vapour pressure to 0.01.3 mm. and the temperature to 0.31°K,but helium still remained fluid. After the death of Onnes, his collaborator and successor, Dr. Keesoni suc- ceeded in 1926 in solidifying helium by subjecting it to an enormous pressure, Helium was compressed in a narrow brass tube under a fressure of 130 atmospheres, the tube itself being immersed in a [quid helium bath. Jt was found that the tube was blocked indicat- • that part of the gas had solidified. If the pressure was reduced I or 2 atmospheres, the tube became clear again. Later expen- ds showed that helium at 4.2*K solidified at 140 atmospheres while at 1.1 °K it solidified only under 23 atmospheres. Solid helium i be distinguished from the liquid : it is a transparent mass having almost the same refractive index as the liquid. 27. Cooling produced by Adiabatic Demagnetisation. — Upto 1925 tiethod available for producing temperature lower th is the boiling of liquid helium under reduced pressure. mi in this way reached 0.72°K in 1932, In 192G Debye and u theoretically that lower temperatures could be pro- by the adiabatic demagnetisation of paramagnetic substances (i.e. those substan ces for which the magnetic susceptibility y ' i i . The principle of the method is as follows ; — M8 PRODUCTION OF LOW TEMPERATURES [CHAP. The process of magnetising a substance involves doing work on it in aligning the elementary magnets in the direction of the external field. If a substance already maguc sei demagnetised adia- batically, it has to do work and the energy to do this work is drawn from within itself, in consequence of which it cools, This cooling can be made large if a .strong magnetic held is employed and the initial temperature is low because then the magnetisation produced in the substance is large. This follows from Curie';; law. which states that the paramagnetic susceptibility of a substance varies inversely as i u absolute temperature i.e. % = C/l\ The final temperature attain- ed is determined by measuring the magnetic susceptibility of the sub- stance and calculating from Curie's law. In this way de Haas and Wiersma succeeded in reaching the present low temperature record of about (UK)34° K in 198 > idiabatkally demagnetising mixed crystals of chromium-potaawum alum and aluminium-potassium alum at' 1.29°K from an initial field of 24000 gauss. 28. Properties of substances at liquid helium temperatures. — Properties of substances undergo very interesting changes at ex- tremely low temperatures. K. Onnes found in 1911 that at abom metals appear to lose completely their electrical i and become superconducting. The resistance does not absolute vanish bin falls bo aboui a millionth of its value so that if a current. be induced in a toil ol Lhe metal placed inside such a low ten bath by bringing a magnet near it, the current does not in die out as in ordinary ele« ti aetic induction, but may continue to together. It. has also been found thai near the absi to vanish. Besides, liquid helium about 2 esses strange properties, the most im- portant being the property of superfluidity when the liquid has no viscosity. 29. Low Temperature Technique* — Dewar ft-.i- The discovery of Dewar flask by Sir James Dewar* in the Royal institution of Loi provided a very convenient apparatus for low tempera- ture storage. Though very low temperatures had been produced, it was d:: to maintain the liquids at these temperatures as even by packing -the bottles with the I: material, the leakage oE heat, from outside could not be prevented. But the problem • ed by Dewar in a very ingenious way, * James Dewar (1842-1523), born at Kincardine, became Pn iess c oJ b Philosophy at Cambridge In 18/5 n,nd also Pi • n the Royal Institution in 1877, His i rfc was in the low temperature region, 15,— Dewar Flask, VI.] USES OF LIQUID AIR 149 The Dewar flask (shown in Fig. 15 together with a siphon) con- sists of a double-walled glass vessel, the inside walls being silvered. The air is completely evacuated from the interspace between the walls which is then sealeti If some substance be now placed inside such a vessel and the top closed, it is perfectly heat insulated, except for the small amount of heat which may creep in by conduction alone the sides. The silver coatings protect the inside from radiation, and the absence of air prevents the passage of heat by conduction through the walls. If the substance be hung by thin wires inside the flask and i,l le latter evacuated by pump and sealed, the insulation is com- plete. Such an arrangement was used by Nernst in his low tem- perature calorimetry (p. 42) . Dewar flasks are now sold in the market under the trade name "Thermos flask". They have lately been made entirely of metal with a long neck of some badly con- ducting alloy as (ierman silver. Low Temperature Siphons. — For transferring liquid air from one vessel to another, special types of siphons are used. One such siphon is shown in Fig. 15, connected to the Dewar flask. It is formed of a double-walled tube silvered inside, the space between the walls being evacuated. On the application of gentle pri • -sure to the rubber compressors A or B, liquid air rises up trie siphon and can be trans- ferred to a second vessel. tvyostats. — For low temperature work constant temperature baths are necessary; they are called cryostats. The substance to be investigated is kept immersed 1 in these baths. From Table 3, (p. 136) it is easy to find out which Liquids are suitable in a particular range of temperatures. In this way suitable liquid baths can be easily constructed down to ~218°C/ When no suitable liquids are avail- able, vapours of liquids can be employed. 30. Uses of Liquid Air and Other Liquefied Gases. — The import- ance of liquid air is being increasingly felt so much so that it has now become essential for several purposes. Bottles of liquid air can now be obtained in any important modern town at a compara- i ill cost. We shall give some of the important uses to which liquid air has been put r (') Prod of High Vacuum, — High vacuum can be obtained by using Liquefied gases with or without charcoal. For instance, if a 1 1 1 fust fdled with a less volatile gas than air, say sulphurous i' id h water vapour, and is then surrounded by liquid air all the gas inside becomes solidified and thus high vacuum is produced. If the I contains air, liquid hydrogen may be employed to condense ii. I Ilia process is greatly assisted by charcoal which possesses the rem property of occluding gases at very low temperatures and the |i ; temperature the greater is the adsorption. Further also •he ion is selective; as a general rule it may be said that the more vola lie the gas, the less it is adsorbed. shall give a numerical example. During a certain experi- i' i in a vessel containing air at a pressure of 1.7 mm. at 15 a C, when l'UODUCTIOX OF LOW TEMPERA!!: •:.;• r.i-.\'. : - cooled by charcoal immersed in liquid air, gave a pressure of 0.000047 mm. in an hour and using liquid hydrogen as the i oak sure was reduced to 0.0000058 mm. (ii) Analytical Uses of Air*— Liquid air is of great use in drying and purifying gases. Water vapour and the less volatile impurities are easily removed by surrounding- the gas in question (say H->) with liquid air, and for this purpose it is now used as a common labora- tory rea,:y (Hi) Preparation of Gases from Liquid Alr.~ Oxygt a k now pre- pared commercially from liquid air by fractional distillation. Since the boiling point of nitrogen is - 195,8 C C, and that of oxygen is - 182,9°*!, the fraction to evaporate first will be rich in nitrogen while -thai evaporating last will be rich in oxygen. A few fractional distilla- tions will suffice to separate these completely , Several r v been devised to effect this separation. In Linda's rectifier (1902) liquid air trickles down a rectifying column where it meet, i upgoing stream of gas. The temperature at the top of the column is slightly below -194°C, {B, P. of liquid air) while at the bottom it is - 183 °C. (B. P. of oxygen). 'Ihe rising gas at the bottom comes in contact with the down-coming liquid and thereby some oxygen of tire risinu condensed, while some of the nitrogen in the downcoming liquid evaporates, and the liquid also becomes oner. The process continues till the liquid reaches the bottom whi mains nearly pure oxygen, while nitrogen passes off as mi at the top. gen is almost pure but die nitrogen con- of oxyg <re efficient rectifiers have since been devised by other workers. For details see Separation of Gases by . Run em an n, Chaps, VI, VII and VIII. gain atmospheric air may be utilised for the production of the rare gases, particularly helium, neon and argon. Roughly five volumes of helium are found in million volumes of air but this is sufficient for our purpose. Liquid air may be separated into two fractions, the less volatile part consisting of O a , N 2 , A, C0 2 , Kr, Xe and the more volatile part consisting of He, H 2f Ne.^ Thus in the rectifier described above the gas going up will contain N tt , LL, He and "N T e. The nitrogen is removed by passing the gas through a dephlegmator and hydrogen is removed by sparking with oxygen. Neon and helium can be separated by cooling the mixture with liquid hydrogen. Thus, oxygen, nitrogen and helium may be obtained from . For details see Ruhcrnann, Separation of Gases, Chap. IX. (iv) Calorimelric Application*. — Dcwar constructed calorimeti of liquid air, oxygen and hydrogen. lie employed pure lead as the heater, and the volume of the gas evaporated by the application of this heat was measured. These calorimeters have the advantage that a large quantity of gas is formed which makes it possible to detect as. little as 1/S0O calories with liquid hydrogen. In this way V1 1 PRINCIPLES OF AIR CONDITIONING the specific heat of lead and other substances may be investigated at low temperatures. (v) Use of Liquid Gases in Scientific Research^-Jhe extreme y low temperatures which are now available to us by the use o£ liquid and ifquid hydrogen have opened for the investigator a new and vast field for research. Tins has made a liquid air plant essentia ,,,;,, modern laboratory. Most of the important OT^""* matter have been investigated at low temperatures and have y, elded X of far-reaching importance. This has been extended even » biological research, where it has been shown that bacteria as well as their activity unimpaired even after exposure to liquid air temperatures though a moderately high temperature is fatal. (vi) Industrial Uses of Liquid G««s.— Liquid air is used com- mercially for the preparation of liquid oxygen as explained above. For submarines anS aeroplanes it may be found useful to store liquid air or liquid oxygen for respiration but the low thermodynamic effi- ciency inherent in the Uncle machine prevents the use of liquid air oxygen "is, employed on a small scale for preparing expk with powdered charcoal and detonated, it explodes with gieat violence. 31. Principles of Air-conditioning.- The Comfort Ctort^-The casonal variations of temperature, humidity, etc, have marked effect m PTowth, longevity and working efficiency of man. The seasonal of the Near lead lis to change periodically our clothing, food on gro changes of the year lead . and manner of living. But we can hardly cope adequately with the variations unless we" can really control the weather changes within our comfort limits as regards temperature, humidity and other factors. The science of refrigeration, heating and ventilating devices have rendered it possible to control the weather at least within the tour wails of our room. This particular branch of study is known as the science of air-conditioning. Complete air-conditioning means the control of the following rs : . .Average comfort condition. 75-7 7 'T. 60-65%. 25 75 ft./min. at least 25% of total circulation. I) Temperature Relative Humidity Air movement Introducing Fresh Air Purification of Air i ii ido tzing [<<iv Li i >r Activating the Air. Although apparently temperature seems to be the only guiding , i,„ ; Lng, the relative humidity (r.h.) plays almost ly imp 'le in the feeling of warmth. The same tempera- m PRODUCrriON OF low temperature? [chap. ture condition, say 75°F, may make us feel either a bit too warm or too chilly according as the r.h, is too high or too low. This is because the humidity condition controls the evaporation from our body and hence the abstraction of latent heat which gives rise to the different warmth feeling. It is interesting- to note thai, some of the air-conditioning plants in the tropics (Calcutta) do not employ any heating device during winter but only humidify the atmosphere by atomised spray of water. This is because the average winter temperature of the place (about 70 — 75°F indoors) is not 'really too low, but what makes us feel chilly is the low (40%) r.h. So we feci quite comfortable only by raising the r.h. up to 60— 70%. In cold countries, however, rooms are conditioned by electrical or steam-pipe heating devices* it is interesting- to find that the comfort feeling is fairly critical, that is to say, that individual variation does not go much oft' the average. Elaborate experiment? have been performed by the Harvard School of Public Health in collaboration with the American Society of Heating and Ventilating Engineers on the average comfort feeling under the different combinations of temperature, humidity and air Dlty BULB TEMPERATURE Fijr, 16.— The comfort chart. velocities, etc. As a result of these experiments the comfort chart (Fig. 16) is drawn in which the co-ordinates are tbe dry and the wet bulb temperatures and lines of constant r.h. and comfort scales for vi.j PRINCIPLES OF AIR-CONDI HOKING 153 cliU summer and winter are also drawn. The chart shows that 98% ol the people during- summer would feel very comfortable at 71°F. effective temperature. We shall see presently what effective tem- perature really means. It relates to the human feeling of warmth under various combinations of temperature, r.h. and air velocity. For example, this 7l°F effective temperature., may be obtained by various combinations, such as 40% r.h., 78°F dry bulb, 62°F wet bulb ; or 60% r.h., 75°F dry bulb, 60 D F wet bulb ; or 70% r.h., 74°F dry bulb, 67 b F wet. bulb (all with air velocities 15—25 ft./min.) . So we see that the term 'effective temperature' represents a new scale which enables us to standardise the comfort feeling due to the various combinations of dry bulb and wet bulb temperatures (which automa- ticalfv define r.h.) and air movements. The three combinations (starting with 40%., 60% and 70% r.h.) as exemplified above would give rise to a feeling as if the person were placed at a temperature of 71°F in a saturated atmosphere and with still air. This is how we can define effective temperature. The effective temperature scale thus represents the conditions of equal warmth feeling with various combinations of temperature, humidity and air move- ^ men is. These are find- ings based on experi- ments with men and women with normal clothing- and activity and subjected to condi- tioned atmospheres of the various combina- tions of temperature, r.h., and air movements. Fig. 17 shows the el i :: c rive tempera ture chart. Let us see how to read the chart to find the effective tempera- ture. Suppose that we have in trie room an atmosphere with dry bulb temperature 76°F, and wet bulb 62 D F and wind velocity "30 ft./ min r (0-30 ft./min, may be taken as good as still air.) Now put the rht edge of a scale across the two temperatures (the dotted line) , and it intersects the effective temperature lines at about 70°F. Tf the same temperature of 7G°F dry bulb and 62°F wet bulb are found to exist with a wind velocity of 200 ft./min. our feeling would be corres- ponding to (574-°F effective temperature and so on." We have seen that temperature and humidity are the most important factors for comfort feeling, while movement of air gives 17.— Effective temperature chart. PRODUCTION OF LOW TEMPERATURES CHAP. a feeling of ease if ic has velocities within 25-75 Et/min. If the air is dead still, it becomes uncomfortable and stuffy. On the other hand, if it has too high a velocity it. becomes blasty and we would not like it. The higher the wind velocity, the colder is our feeling, as it facilitates evaporation from our body. We next consider fresh air. A room-cooler (or a room-heater in the cold season) must make provision for introducing sufficient amount of fresh air into the room. Of the total circulation a mini- mum of 25% fresh air is recommended, the remaining part is the room air itself, recirculating through the machine. It h true that it more air is drawn from outside and cooled in the air-conditioning machine for distribution in the room it would be better, but it becomes too expensive since much more work has got to be done in ord cool the large bulk of outside hot air. In a complete air-conditioning outfit, devices are included to purify and deodorize the air by suitable means. In spite of the complete air-conditioning arrangements it is found that we can never feel like the natural atmosphere under the same conditions. Recently it has been found that the amount of electri- cally charge ions present in fresh atmosphere is higher than that present in the air of an occupied room. Experiments have been made to ionise air by X-rays and introduce the ions into the room in portions to activate the atmosphere of the room. This has given positive effect. 32. The Air-Conditioning Machine, — We have so far seen what iing actually in: Ve shall now consider how it is ved. Fig. 18.— The Evaporator. The air-conditioning machine or the room-cooler is fundamentally a refrigerating machine which has been described in section 7, p. 130 with the main difference in the design of its evaporators. This is THE AlR-CONDlTlONlNG MACHINE 155. VI.] just tl»e question of how we want to utilize the cold produced by the refrigerating machine. In air-conditioning unit the evaporator con- sists of a scries of zigzag copper-tubings thoroughly firmed with thin- copper sheets in order to get a large area of cold surface (Fig. 18) - As the liquefied refrigerant (SO*, Freon, etc.) evaporates in the tubing fier, etc, clearly HB& i**Wl«'«B»r.-*i.v:.. Fig. 19. — Frigidaire conditioner, It is important, to note that die summer air is laden with much moisture and it is desirable that humidity should be lowered. As the air is fanned through the cold fins of the evaporator the moisture condenses on them into droplets which are ultimately drained off. Thus cooling and dehumidification are simultaneously brought about in die same process. 156 PRODUCTION OF LOW TEMPERATURES [CH IP. The size and capacity of an air-conditioning machine is not determined only by the size of a room. It depends' upon the follow- ing considerations of heat loads ; (i) Sun's rays falling on walls or roof. (ii) Conduction through walls and roois due to the difference of outside and inside temperatures, (tti) Human occupancy. — For small private installations this heat load is not more than 5% of the total load but in cinema or theatre halls, it is 55 to 65% and in restaurant 40 to 60%. (Average heat dissipation is taken to be 400 13. Lu. per hour by each person) . (iv) Infiltration, Le. } outside unconditioned air entering through conditioning machine itself (its motor) , cooking stove, etc. (v) Heat-producing items in the room, e.g. tie lamps, air- conditioning machine itself (its motor), cooking stove, etc. In order to minimise the heat load which mostly enters from outside, the walls and ceiling must be covered with insulating boards such as celotcx, masonite, etc. and matting for the floor should he used. America and most of the European countries, But for tropical like India, the chart would differ considerably. People in the tropi omed to more warmth and humid atmosphere, an{1 tni ican air-conditioning- machines have got to readjusted according to our comfort conditions. Books Recommend* 1. Andradi Z I,. C. Jackson, Low Temperature Physics (1950), Mediuen & Co. 3. M. and B. Ruhemann, Low Temperature Physics (l°-§7) , Gam bridge University Press. 4. C. F. Squire, Low Temperature Physic* (1953) , McGraw-Hill Book Co, 5. M. Ruhemann, The Separation of Gases (1940) , Clarendon Press, Oxford fi. Glaze brook, A Dictionary of Applied Physics, Vol 1, articles on 'Refrigeration' and 'Liquefaction'. 7, Mover and Fittz, Refrigeration, 8, Hull, Household Refrigeration, Published bv Nickerson & Collins Co., Chicago. 9, K. Mendelssohn, Cryophysics, (I960), Intersciencc Publishers, Inc., New York. CHAP PER VII THERMAL EXPANSION 1, The size of all material bodies changes on being heated. In the majority of cases, the size increases with rise in temperature, the important exceptions being- water and some aqueous solutions in the range to 4 fi C, and the iodide Of silver (resolidified) below 142°C. We shall first consider the expansion of solids. EXPANSION OF SOLIDS 2, The cubical expansion of solids is somewhat difficult to measure directly (a method is given in section 17) , and is generally calculated from the linear expansion. Hence, experiments on die expansion of solids generally consist in measuring the linear expansion of bars or rods of the solid. For isotropic bodies whose properties are the same in all directions, the expansion is also the same in all directions. To this class belong amorphous solids (e.g., glass) and regular systems of crystals (e.g., rock salt). Metals may also be in- cluded because though they are composed of a very large number of small crystals, these crystals are oriented at random and the average properties are independent of direction. In anisotropic bodies such as many crystals, the expansion is different in different directions and may be even of different sign. We shall first consider isotropic bodies. ISOTROPIC SOLIDS 3. Linear Expansion. — If a bar of length I, } at 0°C. occupies a length I: when raised to t a C. t l t can always be expressed by a rela- tion of the form where X is called the mean coefficient of linear expansion between and t°Q., and is a very small quantity. This differs very little from the true coefficient of linear expansion « at the temperature .' which is equal to 1 dl I dt' The true coefficient a may also be defined ] <!l by the relation a=-, — j~ which on integration will yield (1) if to at fadi — \l. The mean coefficient may be put equal to the coefficient of expansion at f/2°C if the range of temperature is small. Often the initial length is measured not at 0°C but at r_ L Q C. Then if t. z denotes the other temperature at which the lengh is l& we have 158 THERMAL EXPANSION [QIAP. Hence by the binomial expansion A = 5S=S approx The mean coefficient A itself is found to vary with temperature. This implies that the relation connecting- length and temperature is not a linear one and equation (1) must be modified into -where the successive coefficients go on decreasing rapidly. An equa- tion of this type is entirely empirical. The molecular theory of matter lias not yet been developed sufficiently to yield an exact 'theoretical formula. Generally it is sufficient to include terms up to the square of t ; the relation then becomes parabolic. In most cases both the coefficients Aj and Aa are positive, the body becoming more expansible as the temperature rises. 4. Earlier Measurements of Linear Expansion. — The Linear expan- sion of solids is very small ; a bar of iron one metre long when heated from to 100°C, increases in length by about 1.2 mm." To measure such small changes in length accurately, special devices are necessary. The increase in length may be obtained from the readings of a sphero- m eter, or directly observed by means of a microscope. Again , the expansion may be multiplied in a known ratio by utilising the principle of the lever. The most satisfactory method, however, consists in utilising the interference fringes, which is considered in detail later. In this section, we shall consider the earlier experiments. The spheroiriLtcr or a micrometer screw was generally employed to measure the expansion and is suitable for ordinary work. The f. about a metre long, has its one end pressed against ted screw while the other end is free to expand. There is a micro- ■ ■ spheroineter which can be brought into contact with his em noting the micrometer readings when the screw is in contact at 0°C., and at any other temperature (°C„ the expansion of the rod is found, whence the mean coefficient of linear expansion can be calculated from equation (1), Roy and Ramsden employed microscopes to measure the cxpan- □ and were able to obtain results of considerable accuracy. The experimental bar was placed horizontally in a trough between two standard bars, and parallel to them. One standard bar carries a cross-rnark at each end while the other carries at either end an eyi piece provided with cross-wires. The experimental bar carries an object glass at both ends so that the eye-piece on tire standard rod and the object glass on the experimental bar together formed a microscope focussed on the cross-mark on the second standard b: , The standard bars were always kept in ice. One end of the experi- ntal bar was fixed while the other end was free to move when the bar was heated. The object glass was brought back to its initial position by a fine micrometer screw, whose initial and final readings the expansion. ru.J COMPARATOR METHOD 159 1 . — Apparatus of Lavoisier. Laplace and Lavoisier employed the lever principle to magnify the expansion (mechanical lever method) , The change in length was converted into a change in angle by means of a lever arrangement and the angular change was measured by a scale and a mirror or telescope. The principle of their appara- tus is indicated in Fig. 1. One end A of the experi- mental bar AB is fixed while the other end B pushes against a vertical lever OB attached at right angles to the axis Of a telescope LL, which is itself pivoted at O and is focussed on a distant vertical scale CC'. The bar AB is first placed in melting ice and the scale division C ■ en duo ugh the telescope is noted. Next the bar is enclosed in a hot-water bath. The rod AB expands to B' thereby tilting the telescope LOL to the position L'OL' and the scale division C is now seen through the telescope. The expansion BB' is equal to OB tan L BOB' = OB tan ./ COC = OB >( CC'/QC.. Paschen employed a combination of the micrometer screw and the lever. The expansion was multiplied in the ratio 1:5 by the lever and this magnified change in length was measured by the micrometer screw. An optical lever arrangement is also sometimes used when the expansion causes a plane mirror to tilt and thereby deflect a ray reflected from the mirror. 5. Standard Methods. — At the present time the standard methods employed for measuring expansion are ; (1) Comparator Method, (2) Henning's Tube Method, (3) Me- thod of Interference Fringes, Methods (1) and (3) are direct while {2) is indirect. 6. Comparator Method. — This is a standard precision method for ■determining the expansion of materials in the form of a bar or tube. The bar, about a metre long, is mounted horizontally in a double- walled trough so that it can expand freely at both ends (Fig. 2) and has two fine marks L, L made near the ends. A standard metre is i mounted horizontally in another double-walled trough, both these troughs being arranged parallel to each other and mounted on rail: it either "the experimental bar or the standard metre can he >ughl into the field of view of two vertical microscopes M, M. The mi< . are provided with an eyepiece micrometer or can be moved parallel to the direction of expansion by means of a micro- meter screw, and are fixed vertically in rigid horizontal supports pro- jecting out from two massive pillars, the distance between the micro- scopes being about one metre. 160 THERMAL EXPANSION [CHAP, First the two troughs are filled with water surrounded by melting ice in the space between the double walls. The experimental bar is then wheeled into position and the fine marks on it arc viewed Fig. 2, — The Comparator method- through the two microscopes and their positions are noted in the micrometer. The standard metre is then brought below the micro- scopes and the two extreme marks of graduation are viewed through the microscopes. Front the change in the micrometer reading die length o£ the bar at 0°C is obtained. The experimental liar is then heated b\ replacing the melting ice in the double- walled space of the trough by water which is heated under thermostatic control. The fine marks are i through the micrometer eyepiece. Hie increase in length is determined from the change in micrometer reading. For measure- mem •. temperatures the experimental rod is placed in a tube which is immersed in a suitable liquid bath (e.g. liquid air) . 7, Hearing's Tube Method of Measuring Relative Expansion.— In this method the ex- perimental and the comparison bodies are together brought to the same temperature and the differential change of their lengths is measured. The comparison "body is so chosen that its expan- sion in the temperature region is accurately known and, if possible, is also very small. Fused silica serves this purpose well. Inside a long vertical tube made of some well-defined glass (fused silica, Jena glass) , there is a ground point, molten and drawn out of the same glass at. its lower end. Upon this point rests the experimental ind R (Fig, 3) , about 50 cm. long and having both of its end faces ground plane. Upon the upper surface of R rests a pointed end of another glass rod S made of the same glass as the outer wider tube. To the upper end of this rod as well as of the S — _ worn. ft' C- ; i fM t *«*♦ Fig, 3, — Hetmiog's apparatus. outer tube are attached end-pieces carrying scales. The whole tube vn. FIZEAU S INTERFERENCE METHOD 161 up to half of the height of die rod S is immersed in a hot or cold bath and the relative shift, of the end-pieces is measured with a microscope provided with a micrometer eyepiece. The shift gives the relative expansion of the experimental rod against a glass tube of equal length. This is so on the assumption that the temperature of the rod and of the outer tube is the same at the same height. For high and low temperatures suitable baths may be employed. g. Fizeau's Interference Method. — Fizeau devised an optical method '.'< .; upon the observation of interference fringes. This method is capable of very great accuracy and is specially suitable when small specimens of the experimental substance are available, as in the case of crystals* In his original experiment, Fizeau used the substance B (Fig, 4) in the form of a slab about 1 cm, thick with two of its opposite plane X' t -..: M Fig. 4. — Fizeau' s interference method, rail el and polished. It was placed with one of these faces tl on a metal plate A supported by three levelling screws S, S. Vh projected upward through the metal plate a little beyond the upper surface of the slab B. A convex lens L having the Tower ;• large radius of curvature was placed on these screws so tin i in film of air lay between this surface of the lens and the upper polished surface of the slab. With the help of a mirror M and a right ui| I d prism P placed above the lens, horizontal rays of light Erora ;i sodi im fl; ae F were sent down vertically to illuminate the air film and the i ays reflected repeatedly at the surface of the slab and the lower surface of the lens proceeded vertically upwards and were again reflected by the prism P and received by a horizontal telescope T so that Newton's rings could be seen through the telescope. 11 162 THFRMAL EXPANSION CHAP. We know that in the case of Newton's rings the condition lor a bright ring is S = 2 t ie cos r « {In -f- 1) A/2, where 8 is the path difference, e the corresponding thickness of the air film and p its refractive index, r the angle of retraction of the ray into the film, n the order of rings and A the wavelength of the light used. In the present case ^ = 1, r = 0. We have therefore 8 = 2*= (2n -f 1) A/2. .... (4) The difference in the thickness of the film at two successive bright rings is A/2, Hence when the thickness of the film changes due to expansion of the slab and of the three screws supporting the lens the rings appear to pass across a mark in the lens. Since one-tenth of the distance between successive bright rings could be measured, the change in length of the order of A/2 i.e. about 0.00002944 mm. could be determined. When the above arrangement, producing- the air film was enclosed in a chamber which was heated, the thickness of die film changed due to the differential expansion of the screws and the substance E, and the shift oF bright rings across the mark was observed. If x bright rings are thus shifted, the difference in the expansion of the projecting portion of the supporting screw and of the slab B along its cnickna equal to xX/2. In order to find the expansion of the screws* the slab was removed and interference rings were produced by reflections at the surface of the lens and the polished surface of the metal plate through which the screws projected. ►be and Fulfrich improved Fizeau's apparatus by replacing the screws by quartz rings as shown in Fig. .5, p. 163. G and D are two quartz plates and R is a hollow cylindrical tripod, also of quartz, cut with its generating axis parallel to the optic axis, and placed b< G and D. Tin:- specimen is placed inside R and the fringes are formed by ' n-shaped air film enclosed between the lower surface of D and the upper surface of the specimen, the angle of the wedge being very small. The light from a Geissler tube (Fig. G) containing mercury and hydrogen is used. It enters the telescope at right angles, is deviated through a right angle by means of the prisms P, P and then falls upon the system as a parallel beam. The fringe systems for different wavelengths are formed at different heights, in the focal plane of the objective. By turning a screw any of these systems ran be brought in the field of view of the micrometer eyepiece. The lower surface of the upper quartz plate B is provided with a mark of reference and the number o[ fringes crossing this reference mark due to rise of temperature can be measured with die help of the micrometer eyepiece. If X lt A a} A a ,. ...... denote the various wave- lengths of light employed and Xi .-£■ § v .\,-H s , * a +6, the number of interference bands displaced across a fixed line (x representing a whole number and £ a fraction) , the increase A in the thickness of the ah' film is given by VII.] THE FRINGE WIDTH DU.ATOMETRR 168 n : ..- rings of Abbe, Fig. 6. — Apparatus for measuring expansion of crystals by Fizeau's method A= ^(*i+!i) - £ (*+*■)- | (*+*>}■ ^ 9. The Fringe Width Dilatoraoter. — In the last method the change in length was found from observations on displacement of the fringes. Priest devised a dilatomctcr in which changes in length can be obtained from the change in width of the interference fringes. The aratus is indicated in Fig. 1, p. 164. The air film is enclosed between the lower surface of the ro and the upper surface of the base plate, both of which are opti- cally plane and enclose a wedge-shaped space (0.1 to OS mm. thick). ample under test ends at the top in a fine point X xipon which rests the cover plate. On looking down in the direction OO, a system of interference fringes will be seen (as shown in the plan) appearing to lie in the plane bb so that the fringes and the referent and xx on die mirror can be simultaneously focussed. When the sample expands on heating, it tilts the cover plate and thereby thickness of the air film and consequently the width of the fringes'. The number of fringes between the lines ss and xx are observed both initially and finally, and from this the expansion can be calculated. 1G-1 THERMAL EXPANSION CHAF. rlr.:; The calculations can be readily made. We saw from equation (4) in last -section lliat if the film thick- ness increases by A/2, there is a shift o£ one fringe across the mark, the fringes actually contracting. Thus if in the present arrangement the number of fringes between the marks ss and xx changes by x-\-i), and d denotes the distance between ss and and A the wavelength of the light employed, then the change <t> in the angle between the planes bb and cc measured in radians is given by Again, if D is the perpendicular distance from X to knife-edge SS, A the relative expansion of the sample with respect to a piece of equal height made from the material composing the base plate, then ^ is also given by \— Fringe width ditatCHneter. 6) and (7) we get *-*■ £)• CD (8) Knowing A» the coefficient of expansion can be calculated, 10. Discussion of Results*— Table 1 gives die mean coefficient of. expansion of several substances between and 100 D C. The mean coel :!■ multiplied by 10 fi is given in the table, Table 1. Coefficient of Linear Expansion of Substances. Substance Ax 10 6 bstance A >< *0 e pur °C per r C Aluminium : v.;.:-"; Platinum 8,8G Copper 16.66 Palladium 11.04 Cadmium SIM Silver 19.68 Chromium 8.4 '1 ungsten 4.5 Lead 28,0 Brass 18.9 Magnesium 26,0 Invar 0.9 Manganese 22. S Jena glass 8.08 Molybdenum 5.20 Pvrex glass 3.3 ickel 13.0 Quartz glass 0.510 v.q EXPANSION OF ANISOTROPIC BODIES 165 But as already mentioned in section 3, these values change appreciably if the final tempera tore is different from 100 ^C The mean coefficient A is a function of the temperature. As the final temperature is lowered the coefficient decreases. Grimeisen lias found the value of the quantity j ~ for very low temperatures and has deduced an important law connecting die coefficient of expansion and the specific heat. Gruneisen's law states that for a metal the ratio of the coefficient of linear expansion to its specific heat at con- pressure is constant at all temperatures. U. Surface and Volume Expansion,— The change in area and volume can be easily calculated from a knowledge of the coefficient of linear expansion.' A rectangle of sides / and b will, on being heated, have sides of lengths 1(1 +AQ and & (I + At), and its area will become lb (1 -j- Ai) 2 . If the initial and final areas be A and A we have , A , A — A (1 4- 2A1) approx., . . . (9) since A is small. Thus the coefficient of surface expansion is 2 A. Similarly, the coefficient of volume expansion can be shown to 12. Expansion of Silica Glass, Invar.— Silica glass (quartz which has been -lidified into die non-crystalline form) is now : i v esipL or the construction ;of thermometers. The ansion of silica h very small (A = 0.5 X 10 -9 per °C.) , and is very con- ntl) determined by Film's method* Vessels made of this material can be heated without any fear of breaking. The curve connecting the coefficient of expansion and temperature is a straight line ihe room temperature and 1000°C. but at both limits it bends. The coefficient is negative below — 80 C C, Invar is another special substance, being an a!loy of nickel (fCyf ) and steel. Its coefficient of expansion at ordinary temperatures is ex- tremely small and hence it is generally employed for making secon- dary standards of length, and in the manufacture of precision clocks and watches. Anisotropic bodies 13. Tt was first observed by Mitscherlich that the angles be- tween the faces of cleavage^ of a crystal of Iceland spar change when the crystal is heated. He gave the correct explanation of the pheno- menon, viz., that the expansion of the crystal is different in different directions and this is the cause of the change in angle. Such sub- ices are called anisotropic or non-isotropic. For every crystal, however, there can be found three mutually perpendicular directions such that if a cube is cut out of the crystal with its sides parallel to these directions and heated, the angles will remain right ancles though the sides will become unequal. These directions are called the principal axes of dilatation and the coefficients of expansion in those directions are called the principal coefficients 166 THERMAL EX P A MSION [CHAP. of expansion- Denote these by X x , Aj, A^- Then a cube o£ sides U will- on being heated to *°C, become a parallelepiped whose edges will be given by , =y*[l + (A* -M, +**)'] (11) The volume coefficient of expansion is thus A, +*«+**■ The Linear ex- pansion in any other direction can be readily calculated in terjns of the principal coefficients and the direction cosines.* 14. Experimental Methods and Results,— Crystals are best investi- gated by the interference method. The crystal is cut in the manner desired, into a plate with parallel faces from 1 to 10 mm. thick, and is placed between the glass plate and the metal disc. The de- link of these experiments have already been given. When the expansion along the various axes of different crystals is investigated very interesting results are obtained. In the hexago- nal system, for optically negative crystals the expansion along the axis is always greater than that along an axis at right angles to it ; while for optically positive crystals the reverse Is the case. Thus, for Tcelaud spar we have expansion parallel to the axis and contraction perpendicular to it. The contraction is always much less than the expansion so that the volume coefficient remains positive. EXPANSION OF LIQUIDS 15. In case of liquids we have to consider only the cubical ex- am be expressed as a fu anperature ; thus F= P (14-«it |-«,-' : — • )» - • (12) or ap , ,, r«(j *t), ■ ■ ■ ■ (is) ere « is called the mean coefficient of expansion between i t*G. Thus if a mass M of the liquid occupies the volumes F, J ',, at P and 0°C, the densities p, Po of the liquid at the respective temperatures arc p =Mf V, p*= M/V . Using (13) we get the relation " £-.l+«fc / . , . (H) P ** The expansion of liquids is much greater than that of solids, yet il is more difficult to measure, for it is complicated by the expansion of the containing vessel. The expansion _ observed is called the apparent expansion and is a combination of the two effects, '.. expansion of the liquid and of the containing vessel It can be shown (see sec. 17) that the coefficient of absolute expansion of the liquid is approximately equal to the sum of the coefficients of expansion of the containing vessel and the coefficient of apparent expansion of the liquid. Thus the former can be determined if latter two quantities are known, * Further sec Glazebrook, A Dictionary of Applied Physics, Vol. 1, p. 876. WEIGHT THERMOMETER METHOD 167 There are three well known methods for determining the appa- rent or relative expansion :— « B jJ,»TW«if! (i) The Volume Thermometer Method, hi) the Weight Thermo meter Method,' and (Hi) the Hydrostatic Method. 16. The DSatometer or Volume Thermometer Method- The ^ S* divMon of Ltd ! then volumes of the liquid at the two temperatures are ^ + ^ ^ (F# + ^ (I + ^. y being the expansion of the containing vessel. The volume at t°C. is also equal to (F« + *ift) (! + «*>' where « is the coefficient of absolute expansion of the liquid, Equa- 1 ^ehave^ ^ < 1 + -ftsB <r8+% ^ (l + rf> Knowing y the true coefficient « is calculated, or if 7 is not known, the relative expansion «-■ y can be evaluated . 17. The Weight Thermometer Method.-* more accurate method^ deluding upon the determination of weight and not of volume iskmishld by the pyknometer or the weight ttarai^cki. Thtse are vessels so constructed as to take a definite volume of t liquid. The weight thermometer is of the shape shown in Fig. 8, and is made of glass or fused silica. It is first weighed and then completely filled with the liquid by alternate heating and cooling with the open end dipping in a cup of the liquid. The experiment consists in weighing the thermometer filled with the liquid at two tem- peratures. Let it.-!, Wi represent the weights of the liquid filling the thermometer at tern- peratures t x and t 2 respectively. If Vj, F 3 are the volumes of the vessel at the two tempera- tures \md p v p 8 the corresponding densities el die liquid, then w x = F 1PI , u> 2 — Vzp* • (15) if a, y denote the expansion coefficient [quid and the vessel respectively, Fig. 8.— The W i£ Thermometer. ; +vf* l+rh w 2 l-hyr 2 Pj__l+nh Pi l+a/2 1-1 oV . . - ' . (16) —X) ih—h) approx. (17) The apparent expansion a can be obtained from (16) or (17) if 168 THERMAL EXPANSION GH IP, the expansion of glass is disregarded i.e., y is put zero. We then obtain from ; or -S — 1 -}- a{t « — U) app rox. a=* — - L_ approx. (IB) Equation (16) can be breated rigorously. Assuming t 2 to re£er to 0°G, and dropping the suffix I, equation (16) yie w _ 1 ! , BL 1 ! at ' or a = wt w (19) It is thu^ seen that, rij ly speaking, the true expansion coeffi- cient is a little more than the sum of the apparent coefficient and the expansion coefficient of glass, though the difference is almost negligible, and for all practical purposes we can assume a = a-$-y> We could treat equation (10) im rally when the result will be more complicated than (20) . Knowing y the absolute expansion in be calculated. The therm ometi nployed to find the cubical expan- ig the specimen inside the thermo- :r.* 18. Hydrostatic Method (Matthiessen's method), — This consists ill finding the apparent weight of a solid when immersed in the liquid oectivcly. The loss in weight ol: the solid is by Archimedes' principle equal to the weight of a volume of the liq lal to that of the solid; denote this quantity by w. Then where V Xt V% denote the volumes of the solid at the two temperatures t lt f 2 respectively. Then v i _ 1 +y<i Pi_l±al s K " ! l ; and proceeding as before, i ~M? a P 9 1 4- <*V a—y = approx. (21) An equation analogous to (19) can also be deduced. 19. Absolute Expansion of Liquids. — As already mentioned, the three foregoing methods may be employed to find the absolute expansion of a liquid provided the cubical expansion of the contain- ing vessel [or of the immersed solid in § 18] be known. One way * Sec Glazebrook, A Dictionary of Applied Physics, Vol, 1, p vn. HVOROSTAT1C BALANCE METHOD 169 Balance Method. of finding the latter is by calculating it : linear expansion. This is, however, open to objection for Che linear expansion is deter mined from bars of the material and it cannot be assumed a priori that the physical i es of the material do not change when it is an- nealed and worked into a vessel of some shape. For this reason it is best to select vessels of fused silica for which the volume coeffi- cient is extremely small (about 0.0000015 per °C.) . 20. Hydrostatic Balance Method*— There thod of determining the ahsolute expan- i of a liquid which was first given by Dulong and Petit. It depends on the hydro- static balancing of two liquid columns at different temperatures. Dulong and Petit employed a simple U-tube for the purpose. •Regnault brought the upper ends of the tube close together, an improvement which made it easier to observe the difference in height of the two columns. The diagram (Fig. 9) scives to illustrate the principle ol the method. A glass or metal tube, bent as shown in the figure, contains mercury. The vertical columns AB, CD, C'D r are sur- rounded by melting ice and are thereby maintained at (PC, while the column A'B' is surrounded by an oil maintained at any temperature i°G. Suppose that AA' is zontaL Let H, H r t h, h* denote the heights of mercury in the various columns as shown and p, p Q the densities of mercury at t a C, and 0°C. Then since the pressures at D and D' are equal, we have by equating the two expressions Cor the hydrostatic pressure at A, h?p t +H , t> = Hpo+hfr. .... (22) But where c is the coefficient of absolute expansion of mercury. Hence j^+AWf+A, .... (23) whence c can be calculated. If the columns H, h, h f are not at 0°C. but at temperatures , f 3 respectively we shall gee *: + . A ' . _ .jl h t- . . ( , 1+rf ^ l+c^ ' " 1 + ^f, r H-c a '« ' where the quantities c-,, c>, c ? , denote the mean coefficients of expan- sion between the different ranges. These can be determined by having the temperature of A'B' to be £ 1? t%, ( s successively, the height of A' above A is /i la a corresponding term can be adr the right-hand side. Regnault's observations, though carried out with greal must be corrected for various sources of error and hence cannot vi: Id 170 1H BRM A I, EXPANSION CHAP- n rttiti eta ij CH Fm. 10.— Arrangement of Callendar and Moss's apparatus. results of high accuracy, Callendar and Moss repeated the experi- ments aiming at a high degree of accuracy. Instead of a single of hot and cold columns 1.5 m. long employed by Regnault, they wsed six pairs of hot and cold columns each 2 m. long and connected in series as shown diagrammatically in Fig. 10, The hot and cold columns are marked H and C respectively. The differ- ence in height of the first and the last column (viz., &b) is six times that due to a single pair. In the actual apparatus ef, gk , . . were doubled back so that all the columns marked C were one behind the other, and similar was die case with H columns. All the H columns were placed in one limb of a rect- angle and all the C columns in one limb of another rectangle, while the o titer limbs of these rectangles contained electrically heated oil and ice-cooled baths respectively. These were kept circulating by means of an electric motor and their temperatures were determined by a long 'bulb' re tan cc LliiiriTometer, Experiments were performed in the range to 300°G. and an accuracy of 1 in 10,000 was aimed at, 21. Results for Mercury.— Their values for mercury are, howe ifEerent from the mean of earlier investigators such a* ; . Harlow hi made accurate dctermi- i the help of a weight thermometer of silica and aimed at of 1 in 18,000. The concordance o£ results with bulbs of ,ed that as quite isotropic. Th ■ ient of e i in the region to 100°C. is according to Callendar and C. The following numerical examples show how the expansion of rcury Is taken into account in (a) the correction of the barometric ,ture, (b) the correction for the emergent colum of a mercury thermometer and (c) the compensation of the mei m ttm. Example 1 . — A barometer having a steel scale reads 750.0 mm. on a day when the temperature is 20°C. If the scale is correctly 'gradual at o G., find die true pressure, given that the coefficient of linear ex- pansion of steel = 12 X 10~* C C~\ and coefficient of expansion (abso- lute) of mercury = 182X IO - ' 5 per °G The length of scale at 20"C = 75.00 (I + 20 X 1£ X MH") cm. Density of mercury at 20°C ^= p f (1 + 20 X 1S2 X 1{) ~ 6 ) where p Es density of mercury at D C. 75.00(1 4- 20 k 12 x 10^)^ ressurc — dyncs/cm a * (1+20*182x10^ =75.00(1-0.0034) ^=74.745 Ptt g dyncs/cm.* VII.] EXPANSION OF WATER 171 Example 2. — A mercury thermometer, immersed up to S0°C mi in a hot liquid reads 230*0. If the exposed stem has an a temperature of 50°C, calculate the true temperature, given that mean coefficient of expansion of mercury is 182 >< 10-* per n C, and coefficient of linear expansion of glass = 8 X W- 6 D C -1 - Coefficient of apparent expansion of mercury = (182 - 3 X 8)10-* = 158 X 1CM °C^ Hence, the exposed stem, if at the true temperature t*fcl., would occupy a length (230-30) [1 + (£-50) 158 X 10"*]. /. ( - 230 = (230 - 50) (i - 50) 158 X H)-*, when * = 235 U C. Example 3. — F In a mercury pendulum a steel rod of length I cm. D C supports a glass cistern containing mercury, find the height to which mercury should be filled up in the cistern for perfect compen- sation of the pendulum, given that the linear coefficient of expansion of steel a '= 12 X 10- G , linear coefficient of expansion of glass g = 8,5 X 10"*, cubical coefficient of expansion of mercury m = 1.82 >( 10-' I .et h be the required height of mercury in the cistern at C V the volume of that men nd A the cross-sectional area of the at n C. Due to the rise of temperature to t c 'C, the volume ' ' 7 (1 -J- mt) , the cross-sectional area of the cistern increases to A (1 -2gf) and therefore the height of mer- cury increases to Since the centre of gi \\ ic- of mercury at 0°C is at. a height ft/2 from the bottom, it will rise to (ft/2)(l -\-mt-2gt) at t a C, the in- crease being (h/2) (mt — 2gl). The increase in the length of the steel red at t Q C is kit. For perfect compensation these two changes must be equal. Hence imt-2gt) = lot. Oi h =- 2 a 2-2 •'; = _2xl2xl0^_ (182-17) xlCr-* ' tM * J <" 22. Expansion of Wafe*. — Tt is well-known that the expansion of water is anomalous in the region to 4°C, Several workers such as Hope, Despretz, Ma tth lessen, Joule and Play fair and others, measured this expansion carefully. A constant volume dilatometer (sec. 16) may be employed for this purpose. If the dilatometer is made of ordinary glass some mercury is initially put in it to compensate for the expansion of glass. Since the expansion of mercury is 0.000182 and of glass 0.0000255, a volume of mercury equal to due-seventh of the volume of the dilatometer will be required for compensation. These experiments show that when water at 0°C is heated it goes on con- tracting as long- as the temperature is below 4°C. Above 4°(] it 172 THERMAL EXPANSION [CHAP. expands on heating. Accurate experiments by Joule and Playfair show that this temperature of maximum density is 3,95 °C. This anomalous behaviour is usually explained on the assumption that, there exist three types of molecules H 2 0, (H 2 0) 2 , (HgO) 8 , which have different specific volumes and are mixed in different proportions at different temperatures. The total volume occupied is assumed to be the sum of the specific volumes, though there seems to be little justification for such an assumption. PRACTICAL APPLICATIONS OF EXPANSION 23. The expansion of solids and liquids is of great importance in daily life and its consequences have often to be borne in mind carefully. The student will be familiar with most of these from his elementary studies. Thus it is well-known that allowance must be made for expansion in the laying of railway lines and the erecting of .steel bridges, the contraction of metal tyres on cartwheels etc. The expansion of tine steel scale must be taken into account in reading the eter (sec. 2.U Example 1). Similarly the expansion oi mercury must be found out for obtaining the true temperature fro a mercury thermometer (sec. 21, Example 2). Expansion in optical and electrical apparatus also causes difl i. The mirrors of reflecting telesc ich are very accurately figured must be pro- tected from distortion of the ce by expansion of dij . por- different temperatures. The i tee of a coi; on its diirif.r changes with expan- sion con. In dealing with glassware it is particularly Important •i and contraction as the glass is lik to ;:• I to sudden changes of temperature. Thii oor condi heal and the difference in tem- iure between the different parts causes unequal expansion. ;e is avoided by either choosing thin glassware or choosing : ; of small co-efficient of expansion and high thermal con I [vity. Fused si sels ran he heated to white heat and plunged safely into cold water, while pyrex glass can be thrust into a blowpipe flame suddenly without risk of fracture. Platinum wires can be sealed into ordinary glass (lead glass) without any risk of a crack developing because platinum and glass have practically the same coefficient of expansion but a copper wire would cause cracks to develop due to un- equal expansion. 24, Compensation of Clocks and Watches. — For our present pur- pose the pendulum of a clock may be treated as a "simp i . du- rum" with a bob of negligible dimensions suspended at the end of a wire o£ negligible mass. "The time kept hv the clock depends upon the time of oscillation of its pendulum which, in the case of a simple pendulum, varies as the square root of the length or the pendulum. The length of the simple pendulum is the distance between the point support and the centre of gravity of the bob. Thus if this length COMPENSATION OF CLOCKS AND WATCHES 173 vn.j increases in summer due to expansion, the time o£ oscillation will increase and the clock will lose time. If the length decreases due to fall of temperature, the pendulum swings faster and the clock gains. Therefore unless the pendulum is compensated against the effect of expansion the clock will gain in winter and lose in summer. This compensation is brought about by the use of two expansible materials so arranged that the expansion of one is compensated by the expan- sion of the other. In the gridiron pendulum alternate rods of steel and brass are connected as shown h\ Fig, 11 so that the steel rods expand only downwards, and the brass rods only upwards. It is so arranged that 7,cu=ta*2 where l x and l 2 are the total lengths of the steel and brass rods respectively, and «i, og the respective co* efficients of linear expansion. Under these condi- tions the effective length of the pendulum remains constant at all temperatures and the pendulum is compensated. It will be seen that J a /1 2 == W a i ~ 3/2 a pproxi m ate ly, In the mercury pendulum a long steel rod carries, in place o the ordinary bob, a frame fitted with two glass cylinders containing ary. Com- pensation is obtained by the expansion of the rod downwards and the expansion of mercury upwards, (See sec. 21, Example o) . Fig, II, — Gridiron pendulum In watches the rate of movement is governed by the oscillation of a small flywheel called "the balance wheel which is itself controlled by a hairspring. The time of oscillation of the balance wheel depends upon the stiffness of the hair-spring and the moment of inertia oE the wheel about its axis of oscillation. The moment of inertia depends on the diameter of the wheel and is mainly contributed by the three small weights (Fig. 12) screwed on the rim of the wheel. A rise in tempera- ture weakens the hair-spring and increases the dia- meter of the balance wheel. Both these changes cause the oscillations to become slower, the former effect being more important. To effect com] tion, the rim of the balance wheel is made in segments of a bimetallic strip with brass on the out- side and steel on the inside, and weighting the rim with three weights as shown in the figure. Due to the larger expansion of brass the end of each segment curls inwards when the temperature rises thereby reducing the moment of inertia sufficiently to compensate both for radial expansion and weakening of the hair-spring. The precision watches such as chronometers are usually provided with cor i balance wheels. Invar steel is now frequently used in the manufacture Fig. 12.- Compensated balance-wheel. 174 THOtMAL EXPANSION [CHAP. of pendulums and balance wheels on account of its very small coeffi- cient of expansion,. 25. Thermostats. — The property of expansion is often utilised for * tnr 13.— Toluene Thermostat constructing thermostats. In these the temperature of any substance can be kepi constant for a long time. For temperatures upto lGG a C s a toluene thermostat may be used but 'for tempera- tures above JOCC, a bimetallic thermo- regulator is generally used. These will, be described. ' Tolu-ene Thermostat, — A toluene thermostat is shown in Fig. 13. In the bulbs T there is toluene, alcohol or some other liquid having a large co- efficient of expansion. These bulbs are immersed in the bath whose temperature is required to be maintained constant. In case the tem- perature of the baLh increases the toluene expands and forces the mercury (shown black in the figure) up the tube in C and thereby ises the opening leading from A to B. The bath is heated by a burner and the gas ' his hunter is supplied through A and this opening. Thus, on account of the expansion the gas supply is 1 the temperature falls. Tf the temperature of the bath falls too much, toluene contracts, the opening- is increased and By adjusting the amount of mercury in K. the ire can be kept constant at any desired value. In case of trical heating electrical contact can be arranged above the mercury lev:: ft.chmg off the heating current. illic Therma-Regulator. — The Dekhotinsky bimetallic thcr- ulator is shown in Fig. 14. It. consists of a compound strip R. which is made by welding to- gether brass and invar steel. The scrip is wound into a small helix and is connected by means of the rod P to the con- tact maker C which makes or breaks the electrical circuit of the heater H. When the tem- perature rises the expansion of the brass causes the helix to unwind and thereby the con- MAINS tact at C is opened thus break- „...„._,,,,.„. ^ _ . . . . £ , • ., ,,,.,- Fig. 14. — I be Bimetallic Thermo-Rcgulator. :c electrical circuit. With c- the helix contracts and thus contact is made at C thereby completing the electrical circuit. With this apparatus tem- peratures up to 3(K)°C can be maintained to within ±1*C. for a long time. VII. I EXPANSION OF GASES 175 (27) EXPANSION OF GASES 26* Expansion of Gases*— The expansion of gases forms the very basis of the system of thermometry and the perfect gas scale discussed in Chap. I. As stated there the results are best expressed in the form of Charles' law which holds very approximately for the so-called permanent gases in nature. Here we shall describe the experimental methods of determining the coefficient o£ expansion. In the case of gases it is necessary to distinguish be L ween coefficients of expansion : (1) the volume coefficient of expansion « at constant, pressure, and (2) the pressure coefficient of expansion $ ai constant volume. The volume coefficient of expansion is denned as the increase in volume of unit volume at a C for each centigrade : ■■:.?. rise of temperature at constant pressure. Thus <^ = K V J L . (25) where V and F n denote the volumes of a fixed mass of gas at t° and II : C, Or F=F*tl + «9 (26) If the volumes at temperatures t\ and f 2 a* 6 Vi and F 2 respectively* get, with the help of (26). «~ Eta .... We can thus determine a by measuring the volume of a fixed mass of gas at two temperatures. Similarly the pressure coefficient of expansion of a gas is defined as the increase in pressure, expressed as a fraction of the pressure at 0°C. for one centigrade degree rise of temperature when a fb mass of the gas is heated at constant volume. Thus if p and p^ be the pressures at t Q and 0*C we have from which relations analogous to (26) and (27) can be deduced. 27. Experimental determination of the Volume Coefficient of Expan- sion. — Gay-Lussac was among the earliest to measure the volume 'licient accurately. Resmault used an improved form of apparatus and corrected his results for various sources of error. Fig. 11 shows a laboratory arrangement for determining the volume coefficient and employs Regnault's technique in a simplified form. e bulb A is connected by a narrow glass tube to a calibrated limb B of a mercury manometer whose other limb C can be moved up and down for adjusth ry level in. B. The tap T enables the quantity of gas in A to be adjusted. First the bulb A is put in j cold water bath at t t °, and after it has acquired the temperature of the bath, the tube C is adjusted until the mercury stands at the same level in both arms, and the mercury level in B noted. Then the *-- 176 THERMAL EXPANSION [CHAP* is heated, when the enclosed air expands pushing the mercury down in B and up in C, The bath is maintained at a certain tempe- rature and the tube C lowered to bring the mercury level at the same height in B and C, and Uie volume of gas in B read from the graduations. The process is repeated for every 20° rise of temperature up to I00 c C, and the observed readings are utilised for cal- culating a from (27) . The results are hest treated by plotting the observed volumes against temperature on a graph* It is found that all the points lie on a straight line showing that equal changes in tem- perature lead to equal changes in volume at constant pressure. This is Charles' law which may be formally stated : for a fixed mass of gas heated at constant pressure, the volume in- creases by a constant fraction of the ■me at 0°C for each cem . ree rise in temperature. This also follows from the result that a comes out to be the same whatever values of 2 are utilised in (27), Results further show that a = 1/273 (nearly! ...'.. Hed permanent gases For accurate work ecu )e applied for the following sou, : — (1) the gas in the ow tube and the manometer is at a different i emperature From the bath, on of the glass bulb with rise of tern re. Regnault appli erections for these and found that all real gases showed small departures from uniform expansion and that the coefficient of expansion red slightly from one gas to another. 28. Experimental determination of the Pressure Coefficient of Expan- sion.— -The pressure coefficient can be y determined in the laboratory with tiie help of an apparatus known as Tory's apparatus. It consists of a glass bulb A, of about 1 C.C. capacity, which is filled with dry air and is connected by a glass Fig\ 16. — Joly's apparatus for determining pressure coefficient. VIL] PRESSURE COEFFICIENT OF EXPANSION 177 capillary tube to a mercury manometer mounted on a stand (Fig, 16) . A fixed' reference mark X Ls made on the tube B near the top by means of a file, and the mercury level is always brought to this mark by adjusting C before any reading is taken. This ensures that the volume of the enclosed gas is kept constant. A metre scale S is fixed to the vertical stand to read the difference ft in the levels of mercury in the two tubes B and C. The bulb A is immersed in a water bath which is well stirred, and temp.Liai.Lin:- are read with a mercury thermometer. First the experiment is done with cold water in the bath and die difference h m the levels of mercury in the two columns noted. If the barometric height is H, the pressure of the gas is H ±h depend- ; ion whether the level in C is higher or lower than that in B. The bath is dien heated through about 20°, heating stopped, and the bath well stirred. The mercury is again brought to the reference mark and the level watched carefully. When the level becomes steady at die reference mark, the reading- in C is noted. Heating is then resumed and readings are taken in this way at intervals of 2Q\ The j>- (29) coefficient of expansion /3 is then calculated from the relation Pz -Pi P±h—p.: : For accurate work various corrections are necessary. The most difficult to estimate is the "dead space" correction (p. 9) since the exact temperature of the gas in the capillary tube is not known. The expansion of the bulb introduces an error in /S of the order of 1% ; this can be satisfactorily corrected for by adding the coefficient oi cubical expansion of glass to the observed value of p. Experiments shown that p is fairly dose to 1/273 for all the permanent .•.•; which means that for a fixed mass of any gas heated at constant volume, the pressure increases by 1/273 of the pressure at 0°C for each centigrade degree rise in temperature. Accurate experiments however show thai "this expansion is neither uniform for a gas, nor is it exactly the same for all gases. As mentioned in §7, Chap. L these observed deviations of a and j3 from the correct value of 1/273.16 are really due to the deviation of actual gases from Boyle's law. Let a fixed mass of perfect which by definition obeys Boyle's law, have the pressure ad ioe t-'o at 0°C, When it is heated to t°C, the product of pre and volume can be written as jfy>#o (1 + at) or ptft/tt (1 -f- /?r) depending on whether the pressure is kept constant- or the volume is kept cons- Since these products must be equal by Boyle's law, we get a — /3 lor a perfect gas. The expansion coefficient is experimentally found to be 0.0036608 for all gases provided they are reduced to the state of a perfect gas (p— >Q) . "The equality of « for all gases is really a consequence of the kinetic theory. Book Recommended A Dictionary of Applied Physics, Vol 1,. pp. 1. Glazcbrook 872—900. 12 CHAPTER VII! CONDUCTION OF HEAT 1. Methods of Kent Propagation. — When a bar of metal is heated at one end and held at the other by the hand we ■ nsation of heat Heat travels from the hot end along the bur and produces this sensation in the hand. The power of transmitth manner is possessed by all substances to a varying degree and this phenomenon tiled Conduction oj Heat, In the process of Co m, heat in transferred by the actual motion of heated parricli s of matter whether liquid or gaseous. Tins is best illustrated by placing gently some ils of potassium permanganate ;il the bottom of a beaker conn f ainhig water and heating it. Heated water rises up and curls down forming a closed path which is rendered visible by the red colour imparted to the water. In conduction, heat is transferred by 'contact* and there is no apparent transfer or matter. In both the above processes the intervening medium takes an active part in heat propagation, I mi in addition to these there is anoiJi: i i . ii ■■ the intervening medium takes no part, il id near a coal furnace, we feel the sensation of heat. If we ■ the iuzi| we fee] w\ are the source of heat is the e sun, and In the latter case it is at an enormous distance io mate] ial medium b< tween us am M.. This phenomenon is call r'hc processes of conduction and convection arc ./.,- due to the action of the intervening medium while radiatii with the enormous velocit; Eit. In this chaptei aduction of heat. 2. Conductivity of Different Kinds of Matter. — Common observa show that different substances vary enormously in their con- ducting power. A glass rod can be melted in a flame by holding ii at [nt two or three inches away from the Game, while a copper rod under similar conditions becomes too hot to Couch. Copper is thus i better conductor of heat than glass. Metals, in general, are good conductor-, ol heal ; glaj iod and other non-metals are bad con- ductor*, Hold inside a flame two blocks, one of wood and the othev of copper, each covered with paper. The paper covering the copper 1 -I'M i: i , not burnt, for heat is rapidly conducted away h r and mperature does not rise it .nition point. The' paper cover- he wood is burnt, Similarly, wat< be tied in a cup of paper for the; heat, is taken up by water (convection) , Liquids, in general, are worse conductors of heat than solids and are the m \ very simple experiment shows thai water is a bad conductor of heat. Take some water in a test-tube and sink into it a piece of ice by weighting ii. The water at the top can be boiled by Ilea ting it locally while the ice at the bottom docs not even VIII.] DEFINITION OF CONDUCTIVITY 179 melt. Air and gases in general, are even worse conductors than water. Woolen clothing protects us from cold on account of the fact that it contains air in the interstices which renders ir a very bad conductor of heat, The good conductivity of metals is utilised in the construction of the Davy safety lamp. The flame is enclosed in an iron gauze chamber (Fig, 1) and can be taken down a mine. Any combustible gas, if nes in contact with the naked flame and burns inSide the chamber. But the iron gauze con- ducts awa; heat so quickly that the temperature at any point of it does not rise Co the ignition point and the outer gas does not ignite . 3. Definition of Conductivity.— The first, to give a precise definition of conductivity was Fourier who in his memorable Theorie Analytique de la ChaJeur (1822) treated the subject of heat conduction in a masterly way and placed it on firm mathematical basis. We shall first discuss the (low of heal inside a bar heated at one end. Consider a thin wall of the material with u irallel Fa< i . such that heat Hows in a direction perpendicular to the Lftl Ou 0fl be the temperatures of the two faces, I the thickness rea of each lace, then it can be shown experi- mentally that the amount of heat Q flowing through the wall in a time f, when the temperature at every point of tHe bar is steady, is directly proportional to (t) Qi-6 2 > 00 tn e area A of the surface, {Hi) the time t and (iv) inversely proportional to I, that is. Fig; 1, — Davy Safety Lamp. Q,= KA l (1) The coefficient K is a quantity depending upon the nature of 1 1 1 ; iubstance and is called its thermal conductivity. From this relation conductivity may be dqfined as the quantity of heat flowing per tnd through a unit area of plate of unit thickness, when the difference of temperature between the faces is unity. In analogy with the electrical resistance, the inverse of K may be called the distance of a unit cube, \Yi-,v imagine the thickness of the plate to be diminished indefi- nitely. The limitino Milne of * , a or --,- l is — and denotes / / dx the temperature gradient at any point. The minus sign has been used befi n ", because the symbol d always stands for the increase. Hence, the quantity of heat flowing in the positive direction of x in time df across the isothermal surface of area A at any point x is given by d--KA^di. (2) 180 CONDUCTION OF HEAT CHAP. This equation is of fundamental importance in the theory of heat conduction. The Links in which conductivity is measured in the CCS. system are the calorie per second per square centimetre of area lor a temperature gradient of 1 G C. per cm. [cal cm, -1 seer* 1 ''C- 1 ]* CONDUCTIVITY OF METALS 4. We shall now give the various methods of deter mining the thermal conductivity of metak. Methods I to III employ stationary heat-flow while in IV a stationary periodic flow of heat is used. I. Conductivity from Calorimetric Measurement 5. The definition of conductivity from equation (1) provides a simple method of determining the conductivity of a substance. We need have only a slab of the material of known cross-section heated at one end, and measure the amount of heat that flows out at the other end in a known time, as well as the temperature of the two faces. Titus all the other quantities except K in equation (1) are known, and hence K can be evaluated. The method though simple presents considerable experlmem difficulties. It is difficult to measure accurately the temperature of the two faces of a slab of metal. This is best achieved by keeping .embedded in the surface the junction oF a thermocouple, the use of mercury thermometer or resistance thermometer being inconvenient or impossible. Some early experimenters used steam to heat the slab at one end and ice or n cool it at tfoe other end, and assumed ture of the end rare, of the slab was that of steam and water res But if proper precautions be not taken, the od .sometimes gives absurd results This is on account of the fact that a thin film of fluid always at rest, is formed in contact with the si and this has a large temperature gradient. Hence it is essential that we should observe the temperature inside the slab itself by means of thermometers. There are many apparatus based on this method, and one due to Searle is described below. Fig. 2 shows the apparatus diagrammatically. One end B of the rod AB i& enclosed in the steam-chamber S while the other end A projects into another chamber C through which cold water circulates as indicated, the temperatures at entry and exit being T a and T 4 res- pectively. Temperatures at two points along the bar are measured by thermometers T x and T £ ; let these be T x and 7V The whole rod is wrapped round with some non-conducting material Tike worn 1 etc. In the steady state if m grams of water How past the con; of the bar per second, the heat conducted by the bar per second is T —T 1 -r— S- , where d is the distance ?n(Ti—T s ) and this equals &K thermometers T lt T Thus, the conductivity K is between tin: determined, In the foregoing experiment some heat was lost bv radiation from the sides of the bar. This is a source of error and is very vni. CONDUCTIVITY FROM CALORIMETRIC MEASUREMENT 181 easily eliminated if the bar be surrounded with some material at the same temperature as the adjacent portions of the bar. There will be no How of heat perpendicular to the length of the bar as no Fig. 2. — Searle's appara temperature gradient exists in that direction. The surrounding material is called the 'guard-ring'. Beige l utilised this guard-ring method for determining the conductivity of various substances such as copper, iron., brass and mercury. A vertical cylindrical column of mercury is surrounded by an annular ring of mercury. In both the upper surface of mercury is heated by steam while the lower surface rests on a metal plate Thus the temperature gradient as well as the temperatures at the same level in the experimental column and the 'surrounding annular ring are identical! Under these conditions there can be no lateral flow of heat and the annular ring serves as a 'guard-ring'. The lower end of the experimental column projects into die bulb of a Bunsen ice-calorimeter and the heat flowing out at die lower end is found from the indications of the mercury thread. The difference of temperature between several points along the column is determined by four differential thermo-couples. The conductivity K can be calculated from formula (1) . This method has been adopted by a number of workers, notably by Lees, Donaldson, Honda and Simidu, and others. Lees used the rod method for measuring the con- ductivity of many metals throughout the K-f temperature range 18°C to - 170°C The imental rod, about 7 cm. long and 0.58 cm. in diameter, was heated electri- cally at the upper end, its lower end being J£- fixed to the base of a hollow, closed, cylindrical shell, of copper which complete- ly surrounded the rod. The outer cylinder was suspended inside a Dewar flask and immersed in liquid air or heated electrically and thus the desired temperature of experiment -i jl p. Fig. 3. — The two-plate * system. 182 CONDUCTION OF HF.AT CI3AP. was attained. Knowing the electrical energy spent in the heater wire and the temperature at two points on the rod by platinum resistance thermometers, the thermal conductivity can be calculated. Some corrections are however necessary which are difficult to evaluate accurately. Honda and Simidu employed the two plate method shown in Fig'. 3. P|, P 2 are two exactly similar plates made of the experimen- tal material. Between these is symmetrically placed the electrical heater H in which the heat Q is generated. ' For perfect symmetry two cold plates, K, K maintained at the same temperature are placed at the other ends of these plates. Thus the amount of heat flowing- through each plate is Q/2, IL Conductivity from Temperature Measurement — Indirect method 6. Rectilinear Flow of Heal, Mathematical Investigation,— Consider a metal rod heated at. one end. the isothermal surfaces being parallel planes perpendicular to the length of tlie rod, and let the axis o[ x be normal to ihe.se planes. At a distance x from the hot end (Fig. I) let Q be the tempera- ture and — the temperature gradient, t of thickness v at this point. The amount of di) (from equa- tion 2), Qz the heat which leaves the layer at the face x-\-dx is 1 6+dd Qt— Lq ' Heale r o~ - r X 3 a k— Flow of lieat in a rod. heat Q, whid . the lavei per second is -K. dx —KA ii' since $ -; is the temperature of that face. dx Hence the gain of heat by the layer is equal to Now before the steady state is reached this amount of heat raises the temperature of the layer of thickness dx. Let p be die density and c tiie specific heat of the material per unit mass, and let the rate of rise of temperature be denoted by - . The mass of the layer Idx. Hence, neglecting the heat lost by radiation from the surface, we get KA—r-a dx = pAdx. C di JC dW_ _ d*9 pc dx* " dx* - (6) via.] rectilinear elow of heat 183 where h =- K/pc Thus h is equal to the thermal conductivity divided by the thermal capacity per unit volume. This constant k has been called thermal difjttsivity by Kelvin and tkermonwtric conductivity by- Maxwell but the former term is more commonly used. It will be seen that the thermal dilfusivity h represents the change of tem- perature produced in unit volume of the substance by the quantity of would flow in unit time through unit area under unit temperature gradient. Thus for calculations of the rate of rise of temperature the constant k is of greater importance than K as equa- tion (0) shows (as the rate depends not only on K but also on pe) t but in the steady state the rate of flow of heat depends only on K and not on p c (see equation 8). Tf, in addition, the sides of the bar are allowed to lose heat by radiation, this must be taken into account. If E is the emissive power of the surface, p its perimeter and $ the excess* of the tempe- rature of the surface over that of the surroundings, this radiation loss 3 , assuming- Newton's Law of Cooling to hold true, is equal to Epfi dx. Hence we should rewrite (5) as d& Or where KAi-.dx^p Adx.c Tl ax at- m E P pAc' 'litis is the standard Fourier equation for one dimensional flow of heat and any problem in thermal conduction along a rod consist-: simply in the solution of this differential equation. Steady State — A state is said to be steady when the tempera- ture at every point of the rod is stationary, Le. 3 -j - — 0. We have then where ST- 5 * - ««l m' = !i = Kp . h AK (*) If the radiation losses from the sides can be neglected, this equation further reduces to dx* The solution of this differential equation yields = Ax -j- B, ♦ When the radiation toss term is included as in equation (7), 6 must be nuflsared as excess over ike: temperature of the surrouudinjrK. If this term is not included as in equation (6), S may denote the actual temperature or the excess over the surroundings. CONDUCTION OF HEAT CHAP where the constants A and B can be determined from the boundary mnditions. Let these conditions be (1) $ —8 at x = 0, O being the temperature o£ the source ; (2) 9 = 0± at x = I. Then we have ,' ■x, . - where is the temperature at any point x. If radiation losses are not negligible, the solution will be different, Let us assume that in this case & = e" is the solution. Hence, by substitution, n 2 — w 2 , or n = ± m. Therefore, the complete solution is = Ae™ -f Be~™ . , . . (9) where A 3 B are constants, 7. Inge»-Haas56 f s Experiment*— A method of comparing the con- ductivity of different substances based on this solution was employed by Ing» •:' • early as 1789 and is generally shown as a ' l experiment. Bars of different substance?; are coaled with wax and have their one end immersed m a hot bath of oil (Fig. 5) . The wax nielts to different lengths along different bars. Before the steady si is reached the U tire at any point depends on h, i.e., on both the thermal capacity and the thermal conductivity as in;! nation (7). This is why initially the temperature wave is i . red to travel faster along bismuth ii along copper, for the low thermal capacity of the former more than com- pensates for the larger conductivity of latter. When iJi state is attained, however, the wax is It on the copper over a greater length, Let l lf l 2 . . . . . denote the lengths along which the wax has melted on the differer tfc siperature of the bath, measured above that of the surroundings, and 8i, the' temperature of melting wax similarly me;. i the bars are long enough the temperature at their other ends is the same as that of the surroundings, i.e., = 0. The complete solution of this problem is represented by •lion (9). The boundary conditions for all the bars are (i) 0==O at *= co, (it) #=0o at ,v— 0, (tit) 0=0 t &tx=L By substitution in equation (9) condition (i) gives A = G. Con- ';, then Lives R=0 O . Tlie solution then becomes Condition (Hi) then gives tfi-flfr-' (10) or ml = iog,-T— . tus of Ingen-l-j * V)il.] FORBES* METHOD Since log,(V*i) is tfte same £or a]1 the bars we haVC mik = mJ-j — rnJz =s * • • = constant, which from the definition of m implies that / s — J a = s n — -\ — . . . . = constant, (ID provided the different bars have the same cross-section, perimeter and coefficient of emission. Thus K K m i _ (12) Therefore the conductivities are in the simple ratio of the squares of the lengths along which the wax has melted, and if the conducti- vity of one of the bars be known the conductivity of the others can be 'calculated. This is an indirect method. In order to secure the same coefficient of emission the bars are electroplated and polished, & Experiments of Despretz, Wiedemann and Franz. — Desprctz, as early as 1822, compared the conductivity of two substances by n of three temperatures at equal aces. The bars were heated at one end and were provided with a number of equidistant holes ii ( ughout their entire length. These holes contained mercury in which" the bulbs of mercury thermometers were immersed for record- ing the temperatare. The theory of the method can be worked out b the help of equation (9). Wiedemann and Fran/, following- the same principle devised a more accurate apparatus. The bars under test were about half a metre long and mi . En diameter and were electroplated One end of the hair was heated by steam and the remainder surrounded by a constant-temperature jacket. The temperatures at equidistant points were measured by a sliding thermo-couple which could be mani- pulated from outside. Fig. 6. — Forbes' apparatus, Statical exper' ' (The dpttcd curve shows fall of temperature along the bar and tangent to the curve jives the temperature gradient). 186 SUCTION OF HEAT CHAP ill. Conductivity by a Combination of the Steady and Variable Heat Flow 9, Forbes' Method. — One of die earliest, methods of determining- the absolute conductivity of a substance is that due to Fo bes, Though simple in principle., it is exceedingly tedious in practice, Forbes used a bar of wrought iron 8 ft- long and \\ inch square section. One end of the bar (Fig. 6) was heated by being fixed into an iron crucible containing' molten lead or silver, A number of thermometers, with their bulbs immersed in holes drilled into the liar, were employed to indicate its temperature throughout its entire length. After about six hours the temperatures at all points become steady and are read on the thermometers. The temperature distribution is indicated by the dotted line in Fig. 6, and follows the law =&&""* *. T h 'S is called Lhc statical experiment since it deals with the steady state of heat flow. To obtain the heat flowing across a particular cross-section, Forbes determined the amount of heat lost by radiation by the portion of the bar lying between that cross-section and the cold end. These two quantities are obviously equal in the stead since ho heat, flows out from the cold end, this being at the room ture. Forbes achieved it by performing the dynamical experiment* so called because the temperature in this case is changing. For this. purpose a bar only 20 inches long but in all respects exactly similar to th< y'tv was used. First a high uniform teraperati iiinicated to this bar which is then allowed to cooi iii. exact!) the statical bar, and a cooling curve plotted for it (Fig. 7). ; calculate the amount of heat lost by the statical bar from the point x = x- x , to the end of the bar (x = I) . The amount of heat lost per second by radiation from the surface of the bar from x to x + dx in the steady state, is d& = v -}-<i ,U>:, P c w , Fig, 7, — Temperature curves in Forbes' bar, where p is the density of the material, c its specific heat and --j- the rate of cooling of that element. Hence the total heat lost by i he portion of the bar from x = « lt to x = I in a second is * James David Forbes (1809-1868) was Professor of Natural Philosophy in the University of Edinburgh from 1«J3 to i860. VIII.] CONDUCTIVITY BY PERIODIC FLOW METHOD 187 <-j Apc \w dx- This will also be equal to the heat crossing the surface at x = x lf viz., since the bar is long enough so that its other end is at the room tem- perature and hence there is no loss of heat from that end. Equating : i se two quantities we have Kldd P c :d$ -(f) = £*■ (13) de For calculating — the dynamical experiment is performed on an actly similar specimen with the same exposed surface. The observations are plotted in Fig. 7 which is self-explanatory. The 8, x and 0, i curves are drawn from actual observations while the values of— corresponding to various values of are computed from the 0, t curve and plotted as indicated. Equation (1$) yields — tan <p=P, ,■>.- where tan <£ and F are indicated in the figure. The area F of the shaded portion can be measured by means of a planimeter and hence K calculated. Tliere are several, sources of error in Forbes' method and hence the method fails to give accurate results. The specific heat docs not aain constant at different temperatures- as assumed by Forbes, Fur- ther the distribution of temperature inside the bar in the statical and dynamical experiments are different. Forbes" method has been im- proved by Callendar, Nicholson, Griffiths and others. rv. CONDUCTIVITY BY PERIODIC FLOW METHOD 10. Angstrom's Method. — The conductivity' of a metallic bar can also be found by periodically heating and cooling a portion of the bar and observing the temperature at different times at two points along the bar. This method was first employed by Angstrom*. In his early experiments a small portion of the bar was enclosed in a chamber through which steam and water at any temperature could be alternately passed. In later experiments the end instead of the middle was heated. The bar was heated for 12 minutes and cooled also for the same time, the periodic time being 24 minutes, Texnpera- * Anders Jonas Angstrom (1814-1874) was Professor of Physics at Upsafa. He made important researches in heat, magnetism and optics. 188 CONDUCTION OF HEAT [CHAP, mres were observed every minute at two points along the bar by means o£ two thermocouples. The mathematical analysis of this case is somewhat complicated and will not be given here. We have to solve equation (6) such that the solution is a periodic function of time. 11. Conductivity of the Earth's Crust— The periodic flow method is very suitable lor finding the conductivity of the earth's crust. The earth's surface is heated by day and cooled by night. This alternate heating and cooling effect travels into the interior of the earth in the form of a heat wave {diurnal wave) and gives rise to the diurnal Ltiona in temperature at points inside the earth's crust. Again, the earth receives a larger amount o£ heat in summer than in winter and this causes a second heat wave having the period one year (annual wave) which is also propagated into the interior of the earth. Assum- ing the waves to be simple harmonic as a first approximation (the annual wave.., in particular, departs considerably from this ideal state) > : find how they travel into the earth. The problem is thai bar pa iodi ally heated and cooled at one end and provided with a i J - — ty * A simple harmonic solution of (6) is fl = ^-** sin (orf+fi*+y) s . < - (14) which gives the temperature fluctuations at any point x. This is the equation of a damped ogressive wave and is graphically represented in Fig. 8. The wave moves for- ward with the velocity u /5 whil its a ipii- tude on diminish- ing exponentially (given by o ff~**) which is shown by the dotted curve. These fluctua- tions will be superposed on the mean temperature at any point which will also diminish as we go farther from the hot end. Tt. can be easily seen that the wave-length of the temperature ■wave is A = — Sor/jQ. Again, by substitution from (14) in (6) we get —a Hf-st- • • • < 15) Hence a=Jv}kT. Now if a number of thermometers are embedded in the earth at different depths, the progress of the temperature wave inside the earth ran be investigated, and knowing the wave-length, the diffusivity h can be calculated from (15), In the same way we cap find the conductivity of any bar if we heat one end in a simple harmonic manner as was done by King, -cAX 8L— Temperature wave at a particular instant. VIII.] CONDUCTIVITY OF THE EARTH'S CRUST 189 12. Applications,— The above mathematical treatment can be utilised for solving the deal problem, viz., the penetration of the dailv and annual changes of temperature within the earth's crust. It will be easily seen fro I '.: that the range of ajnphtucK R at any point is J2=20«0-V«/A? and velocity of propagation — ?= 2 j/f • ■ ■ < 17 > and wave-length A= 2yV/*T. : • - ( 18 ) The "lag/' i.e, } the time which any temperature applied at the surface talces to travel to a point x is v 2 (/ rt' ,l.i. We shall now apply these results to the propagation of heat in earth's crust. Here for the daily wave T = 24 hours — 86400 sec. Taking h = 0.0049, the value for ordinary moist soil, we have A — 73 cm.; v = 84 X '0 * cm. /sec. ; <* = 1/11.5. Suppose the maximum temperature is 45°G at 2 p.m. and the minimum is 25°C. just after sunrise, the amplitude of temperature variation is, therefore, 20*C This variation will diminish as e ~*iiL.i at a deptli x. For x — 10 cm., *-■/»♦« = .42, for x = 30, tf -*ni.s_ j|-|; an d for x — 100, *-*/"••= .00018. Thus at a depth o!.' 1 metre the temperature variations will be scarcely noticeable. The slow velocity. of penetration of the daily heat wave must hi been familiar to all observant minds in a tropical country. Here the roof of the house is exposed to the scorching heat of the sun. and at 2 p.m. the temperature may be as great as G0°C. But this tempera- ture travels inward at the extremely slow rate of 9,1 X 10~ 4 cm./sec- or 6.4 cm. per hour in a mass of concrete (h — 0,0058) , Hence to penetrate a wall depth of 30 to 40 cm. a period of 5 or 6 hours is necessary. The inside of the room, provided the windows are shut, therefore, reaches its maximum temperature at about 7 or 8 p.m. when the walls become intolerably hot and begin to radiate. Most :. must have experienced that it is found impossible to sleep •rs at this time. The minimum at the top of die room is reached at about sunrise. So the rooms are found to be cool from ..v£. to about 2 p.m. when the outside is hot. Annual Wave. — Besides the diurnal fluctuations the surface of arth is also subject to an annual period of 3G5£ days owing to the different amount of surface heating in winter and summer. This amount is variable in different countries, but in desert countries it may amount to as much as 60°C in. the sun in summer and 0°C. in winter. 190 CONDUCTION OF HEAT [CHAP. v.: ;'. In the case of t he ann ual wave. X = 73V 365 /' 1 — 73 X 19-1 CD1 - = H metres : 8.4 X JG~V^^ 5 c W sec - ~ 3- 9 cm - P er da y- 1_ 220 ' Thus the annual wave will penetrate a depuh o£ about 1.2 metres month, At a depth of x=\/2=7w. the times of the year fl.6 1/3(35 ~2 » interchanged. The amplitude of temperature variation will he reduced by the fraction e~™>™ — ,64 at 1 metre depth and bv fiimt*» = ,011 at 10 metres depth. Thus the annual wave is able to penetrate to a depth which is V365/1 = about 19 times greater than the daily wave. Most of these conclusions have been verified wall is made up experimentally. 13. Conduction through Composite Walls.— If a of a number of slabs of thicknesses x t , x 2 « . . and conductivities &i> ,v joined together, the amount of heat flowing per second through an area A of the wall in the steady state is a- *i E t A . ,;i.re the temperatures of the intermediate surfaces and lr s u are the bemperaturea of Line end faces. This is because in ite the same amount of heat, must flow through each "sice #i * t -0, = (fcjJ 0a~ fl » — Q- f~5 A « I ••■ and by addition b-** . - a ( jg ^ Therefore the heat, (lowing per second is a--: *l_+ 3 (19) + K\A ' K 2 A 14. Relation between the Thermal a ad Electrical Conductivities of Wienemann-Franz Law.— A table of thermal conductivities is given on p. 200. The table shows unmistakably that all good con- ductors of electricity are also good conductors of heat, and even before any theory was proposed, Wiedemann and Franz gave the empirical law that the ratio of thermal and electrical conductivities at a particular temperature is the same for all metals. Lorenz extended die law and showed that this ratio is proportional to the absolute temperature, viz., KfaT = constant, where K is the thermal conductivity and a the electrical conductivity, vm. HEAT CONDUCTION IN THREE DIMENSIONS 39r Drude explained this remarkable result by assuming that the conduction of heat and electricity in metals takes place by means of tree electrons. He even found theoretically the value of K/&T. These conclusions have been experimentally verified by Jaeger and Biesselhorst and by Lees. They obtained die value of K/vT experimentally for various metals which are given in Table L It will be seen that the value of K/qT remains practically contant for all metals. Table L— Values of J~X LO 8 - a i From Lees' experiment Jaeger and l ii: i.sclhorst Metal - 170°C. 1.50 - ioo°c. o°a 17 D C 10°C. 2.19 100°C. Aluminium . - 1,81 2.09 2.1.3 2.27 Goppi 1.85 2.17 2.30 2.32 2.29 2.32 Silver 2.01 2.29 :'. " 2.33 2.36 Zinc 2.20 2.39 2.45 2.43 1 2.3,3 Cadmium 2.39 2.43 2.40 2.39 2.43 2.44 Tin 2.48 2.51 2.49 2.47 2.53 2.49 "J cm: 2.55 2.54 2.53 2.51 2.46 u\ Iron 3.10 2.98 2.97 2.99 2. 70 2.85 Brass 2.78 2.54 2/1.3 2.45 . , . , Manganin 5.94 4.16 3,41 3.34 3.14 2.97 A closer examination of the tabic shows that the value falls off at low temperatures. Now at these temperatures both the thermal and electrical conductivities are found to increase. Hence' it follows that the thermal and the electrical conductivities do pot increase in the same ratio when the temperature falls, the electrical conductivity increasing much more rapidly. In fact, the latter appears to become extremely large at the absolute zero, if the metal is free from impurities. 15. Heat Conduction in Three Dimensions. — Till now we have con- sidered the flow of heat in one direction only, generally along bars great length but small width and thickness. We shall now con- sider the conduction of heat in three dimensions inside an isotropic body. Tt can he shown by an amplification of the ideas given in sec. fi that for flow in three dimensions, the equation of conduction is given by ri&d , d*8 , d*9\ m or- «*-£ (20) 192 CONDUCTION OF HEAT CHAP. where h is the diffusivity. This is the Fourier equation ^ of hear conduction in general and we can solve it when the initial con- ditions are given. Equation (20) is of great importance in &tu dy- ing problems on heat conduction* V, Conductivity of Poorly Conductinc Solids 16, In finding the conductivity of poor conductors the sub- stance cannot be employed in the form of long bars or rods as was the case with metals, for the heat loss from the sides would be considerable compared with the heat actually conducted away through the substance itself. For this reason the substance is generally used in the form of thin plate, sphere or cylinder. To this class belong- all non-metallic bodies. Cork, asbestos, clay, wood, bricks, etc. are among the many substances of common occurrence. The conductivity of these varies from 0,01 to 0.00008. We shall now give some of the important methods of finding the conductivity of these substances. In the methods commonly employed for finding- conductivity of Soorly conducting solids, energy is supplied electrically to a separate eating body, and the flow of heat through the experimental subs- tance investigated. 17. Spherical Shell Method. — The simplest case of heat conduc- tion in three dimensions is that of a sphere. If the source of heat, is iced at the centre of the sphere the isothermal surf; ill be 1 surfaces described about, the centre. The method has been employed ind others. Nfusselt's apparatus is shown in 80 cm- in diameter and in;;. pper. Inside it tcentrk with it, is another hoi! sphere 15 cm, in diameter. The spheres can be split into two ted. The space between die spheres is filled with the material B under test such as asbestos, powdered cork, charcoal etc. An electrically heated body D is placed at the centre of the sphere inside C and electrical energy is supplied at a constant rate. Temperatures are deter- mined by means of thermo-couples :i Eh B at different distances from the centre and alonir one or more radii. Knowing- the elec- trical energy spent and the radii of the shells, the conductivity can be calculated. We shall solve the problem from elementary principles. On 'lit. of the symmetry about the centre the isothermal sua i Clerical. The flux of heat across a spherical surface of radius r outwards in unit time is — KAfffi t-. This must equal the amount d r Fig, 9, — KTosselt's apparatus. VIII. J CYLINDRICAL SHELL METHOD 19-5 of electrical energy (J supplied per second to the heating body. Hence* Or 4 77 A dr dr Since () is independent of r we have on integrating (21) 8 = Q, 4tt/l" 4+* (21) (22) where A is a constant of integration. Tf the two surfaces of the shell of radii r lt r 2 acquire temperatures 8 l3 &£ in the steady state, we have 4irR' r t ^*' * 2 4wK - -4- A, nee 0.= 4fliT(^-fl Jf a Knowi g 0i,02 and r x , r 2 as in the above experiment the con- ductivity K can be calculated. Again, solving- the simultaneous equations for C> and A, and substituting these values of Q. and A in (22) , the Cempeiature distribution in the material can also be found, and is given by the expression « = r 1 r [Mrt«+ ft ~' JV 4 .* ■ (241 18, Cylindrical Shell Method. — Let us consider the radial flow of heat in a cylinder which has an electrically heated wire along its axis. By suitably transforming equation (20) we can get the equation for this case. We shall, however, as before, deduce it from elemen- :>nsideraliom. Since the cylinder is symmetrical about its axis, the isothermal surfaces are cylindrical, The amount of heat {) Rowing per second across an isothermal surface is ft .- -2wrfjr=. p <-.-•• where I is the length of the cylinder. Now (J must remain constant, being equal to the electrical energy supplied per second. On integrat- between the inner and outer' radii of the cylindrical shell with the corresponding- temperatures B it & 2 , we have or 2W" 8 x -0 t (26) The temperature & at a distance r can be shown to be given by 9 = log (rJr ) [ ($1 l0g r *~ $ * l0g ^ _ ^i-^ lo « '") ] ■ • *We can also proceed as in sec. 6. 13 194 CONDUCTION OF HEAT CHA '■ distance from en nation (2,6) II lias devised a method of determining conductivities based on this methi n er o£ the material is formed by filling space between two hollow concentric cylinders with that material. Heat is supplied by a wire carrying an electric current along the axis oE the cylinder, and when the' steady state is reached, temperatures at two points within the material are observed, as well as their the axis. Knowing - the electrical energy spent, gives the conductivity. simple laboratory expc rimeiit based on this method may be devised for finding fch conductivity of rubber and glass in the form of a tube. For rubber the arrangement shown iu Fig. 10 is most convenient. Steam from the boiler A tra- verses through a rubber I H t a length J! of which is mersed in a weighed amount of water contained in the calori- meter C. The radial flow of heat Q_ from the tube to the water in C is given by Fig. 10,- -Apparatus for finding conductivity of rubl log. (28) re $i n the temperature of steam, and {.)-, the mean of the initial and final tern] ■ of the calorimeter. This heat can in temperature of the calorimeter and its id hence equating these two quantities we get it*. For ;emenl shown in Fig. 11 is convenient. is tube C has a spiral wire along its axis and is surrounded by i mi jacket, while a steady current of water flows through the tube.\ The method of calculation is similar to the preceding one. Walt,- Sieam W ate * Fig. lb— Apparatus for finding conductivity a£ glass. 19. Lees* Disc Method* — Another convenient method for finding conductivity of a very bad conductor has been given by Lees ♦For experimental del- Worsnop and Flint, /Vc ■'■. asks. DHL] LEES' DISC METHOD J 95 Steam in which the substance is used in the form of a thin disc. The apparatus (Tig. 12), due to Lees and Charlton, consists of a cylindrical steam chest A, the bottom of which is a thick brass block B in which a hole is bored for inserting a thermometer T lf The substance S (shown shaded), En the form of » a circular disc, is sandwiched between the block B and a second cylindrical brass block C, the latter carrying a second thermometer I s . The I^ck C is suspended m a retort stand by threads so that its top face is horizontal. The radius and the thickness d of the specimen are first measured. Fig. 12— Lees' disc method f f bad conductor. The apparatus is then set up as shown in Fig. 12 and steam is d through A for a pretty lone time until the readings in the thermometers T 3 and T 2 are steady. These temperatures can be received by it from conduction through the specimen must be equal to the rate at which heat is lost by it by radiation from the sides and the bottom surface of C. The former is given bv G-JM&^&, . . . (29) where A' is the required conductivity and A the area of the disc. The latter is determined in the following iva; ; The block C h alone heated separately by a bunscn burner until its temperature rises to about !0°C above 2 . It is then alone sus- pended with the specimen placed on the top and allowed to rool. Its temperature is noted at regular intervals of time until it cools to about 10°C below g 2 and a cooling curve drawn, From the tangent to the curve at 8 2 , the rate, of cooling - f at 0., is determined. The rare of loss of heat by cooling is a— 'GL . (30) where m and s denote respectively die mass and the specific heat of the block a Combining (29) and (30) , K can be found. In die above we have assumed that no heat is lost from the curved surface of the specimen, and consequent! v the specimen must ery thin. The disc method can also be used with electric heating when a modified arrangement is necessary. A thin plate of the experimei] J 96 CONDUCTION OF HEAT substance is placed between two copper plates, A coil o£ insulated wire is place above die tipper copper plate and held down by another copper plate as explained later in § 22. Electrical energy is supplied to the coil and steady state obtained. The amount of heat passing through the experimental disc can be obtained from the electrical energy supplied and the loss of energy due to surface emission. Knowing cue temperature of the copper plates the conductivity be calculated. Alternatively, the two-plate system can be employed by placing above the heater coil another sandwiched experimental plate, exactly similar to the system below the heater coif. In this arrangement it is not necessary to know the radiation loss as all the electrical energy is transmitted through the two specimens. CONDUCTIVITY OF LIQUIDS 20. The determination of the conductivity of a liquid is complicated by the presence of convection currents. If we heat a column of liquid at the bottom, the liquid at the top receives heat both hy conduction and convection. The laws governing- convec- tion currents are complicated] hence it is preferable to eliminate them. This is accomplished either by taking a column of liquid and heating it at the top or by taking a thin film of liquid. 21. Column Method. — The column method was employed by Dcspretz long ago. The liquid at the top was kept heated and tem- peratures axis ot the column were observed by mercury He Found that Fourier's equation already derived for a bar hold? true in this case riiso, and hence the conductivities of two Liquids caj with the help of equation (11). Weber sur- fed the column with a guard-ring and used as the source of heat an electrically heated oil bath. The bottom of the column was cooled copper plate standing in ice. The heat conducted away was found from the amount of ice melted. Knowing the temperature bution along the column the absolute conductivity can be calcu- lated in the same manner as for mercury in Berget's experiment. 22. Film Method,—- This was employed by Lees>, Mihier and Chat Lock and by Jakob, We shall describe the apparatus iiied by Lees which is simply a modification of his disc method for finding the con- ductivity of poor conductors. The liquid L under test is enclosed in an ebonite ring E (Fig. 13) and placed between copper blocks, C^ C.,. To de- termine the quantity of heat flowing through the liquid a glass disc G is inserted, above which another copper plate Above C 3 and insulated from Fig, 13. — Lees 1 apparatus for finding the rouductivity of liquids. C 3 is cemented with a layer of shellac. CONDUCTIVITY OF GASES 197 it by mica is placed a flat spiral coil of heating wire W which is held down by another copper plate C. The whole pile is varnished and enclosed in an air-bath. Temperatures are recorded by thermocouples soldered to the faces of the copper plate. The calculations can be easily made. Let Si, S s , Sg, S a denote the emitting surfaces of CY, C 2! C s and G respectively, h their emissivity and T-,., T„, T & the temperatures i I' the copper discs. The heat passing through the middle section of G is by definition equal toai£T G (T a - F a )/d . Of this the amount 9 ' —^o — 2 i s I° 5t fry tne lower half of G by radiation from the sides and similarly S 3 h T,-. is lost by C s . Hence the heat transmitted through the liquid and the ebonite is equal to AK t T 9 -r t — SqH ' -'— -, — - — $tkT s and thus must evidently be given by : B(T,-T i ), some constant giving the transmission of energy through the ebonite E for unit difference of temperature. Equating these two expressions we get K& in terms of K G provided B is known. B can be determine! ::rking with air whose conductivity is known. 23. Hot-wire Method. — Goldschmidt employed the "hot-wire" method of Andrews and Schleiermaeher, the theory of which is dis- cussed in sec. IS. The liquid was contained in a silver capillary Lube 2 mm. in diameter heated by a wire running along its axis, CONDUCTIVITY OF GASES 24, The determination of the conductivity of gases is difficult for the phenomenon is always accompanied by radiation, and sometimes by convection currents also. Kundt and Warburg showed that the rate of cooling of a thermometer immersed in air remained constant for pressures lying between 150 mm. and 1 mm. Hence the effect of convection currents is negligible in this region and heat is lost only by conduction and radiation. To determine the radiation loss, the air was exhausted as completely as possible, and then the rate of cool- ing was found to be independent of the size of the enclosure, showing diaf the effect of conduction was negligible and heat was lost only by radiation. Subtracting this radiation" loss, we get the heat lost hy conduction alone. For finding the conductivity of a gas at pressures higher than 150 mm. the gas may be exhausted to this pressure (be- tween 150 mm. and. 1 mm.) and its conductivity determined. Now since the conductivity of a gas is independent of the pressure (see Chapter III, Sec. 24) this will give the required conductivity. Another 198 COXDUCriON OF HKAT [CHAP. Fiff. 14. — A '_:■•.: 'd'atus to sllf>w the better conductivity of hydrogen- procedure consists in taking a thin, film of gas and healing it at the top when convection will be absent A very simple experiment described by Andrews and Grove allows qualitatively that, hydro- gen is a far better conductor of iat than any other gas, A fine latinum wire was supported in- side a glass tube -(Fig. 14) which could be filled with any gas. . wire could be heated by an electric current and made to glow. Two such tubes were ; ranged side by side, one filled with air and the other with hydrogen. The same electric: current was allowed to I through both the wires. The wire in the air-tube can be glowing while the wire in the hydrogen tube does not glow at all. The tss tube containing hydrogen also becomes hot. Heat is very quick- ly conducted awa '■■ and hence the wire is not raised to the temperature of incandescence-. Replacing hydrogen with air res- totes the incandescence. [<e important methods of finding the a i xmductivity of a ■ Method, (2) Film Method. 25, Hot-wire Method. — This method, first given by Andn emi. acher lor determining the absolute conducti- vity electrically heated vertical wire is surrounded by a coaxial c filled with the experimental gas. The temp. e of the wn:e is known from its resistance, and the amount of heat flowing across the curved surface of the cylinder is found from the rate of energy supplied to the wire. The conductivity can be calculat- ed from equation (2ft) where r t , r s now denote the radii of the wire and the tube respectively. In this arrangement convection is very much minimised sine:: there is no temperature gradient in the vertical irei ion, Radiation losses and whatever convection losses remain, eliminated as pointed out. in sec. 24. Correction must, however, be applied for loss of heat by thermal conduction along the leads supplying the electrical current. The method has recently yielded ■• accurate and reliable values for a large number of gases. 26, Film Method, — The film method, originally .used by Todd, has been employed by Hereus and Laby. The principle underlying it is th^ same as in Lees' experiments (sec. 22) . The thin film of gas under test was enclosed between two copper plates B and C (Fig, 15), viu.'l CONDUCnvrTY OF GASES 199 the latter of which was cooled by a current of water. The upper plate B was made up of two sheets of copper clamped together with a heat- nl between them. To prevent loss of heat by radiation from the upper surface of B, there was another plate A above it at the same temperature and a guard-ring D surrounding it. On account of the latter the flow of heat from 1> to C was linear. The plates A, B, D separate heating coils and thermocouples and were kept at lame temperature. The thermocouples- were formed by attaching untan wires to the copper plates, each plate having a copper- lead also attached to it. The plates were accurately ground and silver plated. The whole apparatus was made air-tight by a ring of rubber clamped to A and C by steel bands. The temperature of A was gener- Ebanitc BcaiS iV iier . IS.— Apparatus of Hereas and Laby. ally kept a little above that of B in order to eliminate :; ability onvection. This, however, necessitated a small correction. The radiation correction was determined by a separate experiment on a silvered Dewar flask and was only about 5 per cent. Convection effects arc: entirely absent since tlie gas is heated at the top, The energy spent in B was known electrically and subtracting from if the lost by radiation, the heat transmitted to C by conduction through the air film wis found. Knowing the temperature of C the conduc- tivity of the gas can be calculated from equation (1) . 27. Results, — The thermal conductivity of a number of sub- stances is given in Table 2, p. 200. The value of K is given in calorie cm.- 1 sec.-? C- X * It will be seen from the table that silver is the best conductor of heat (K — 1) and copper comes next. The conductivity is less for liquids and least for gases. The conductivity of gases is extremely (of the order of 1CH 3 ) . We have: already considered on p. 85, Chap. Ill, the relation between thermal conductivity and viscosity of gases, as well as the variation of the thermal conductivity of gases with pressure. 200 -lUCTlON OF HI ■ ■■. : [CHAP, Table* 2. — Thermal conducti of different tances. Substance Metals (0°C.) Aluminium , , j . Cadmium . - ! 0.23 Copper 0.93 Iron (pure) 0.16 Lead 0.085 Mercury 0,02$ Nickel 0.1-1 Silver 1.0 Platinum 0,17 Tin 0.155 Zinc i. :.r:.< Asbestos paper Cardboard Cork (p=.16) Paper Ebonite Mica Paraffin wax Pine wood Rubber Alloys (0°C.) Brass an Mauganin 0.05 0.06 Poo\ 1 lint glass 2.5x10- 2 Liquids Water (24°) Alcohol (25 D ) Glycerine (25*) 0.6k 10" 0.5 0.11 0.3 0.42 1.8 0.6 0.4 0.45 14.3XJ0 4 4.3 jes (0°C.) Helium Argon Air <:Ll Carbon dioxide 10 ■* 3.89 5.40 5.63 8.07 28. Freezing of a pond.— An interesting example of conduction across a slab of varying thickness is provided by the phenomenon of freezing- of water in ponds and lakes during winter. When ice I to form on a pond, the bulk of the water in the pond is at aboui while the top layers are at 0°G, and the cold air above abstracts the latent heat from a narrow surface layer at 0°C. The subsequent growth of rile ice layer requires that the necessary abstraction of latent heat take* place by conduction through the layer of ice already there. Assume that, the thickness of the ice layer already formed is z, and die temperature of the water below this layer is 0°C. Then for a further freezing of the layer of area A and thickness dz in the time dt 3 the quantity of beat Q = Apdz.L -Taken partly from Landolt and Bernstein, PhysikalUh-Chemischen Tabelten and partly from Kaye and Laby, Tables of Physical and Chemical Crttwttents, CONVECTION Ol 7 HEAT 201 vii i.] uu-.l travel upwards by conduction through ice of thickness t, wner P = density of ic& and L — the latent heat of fusion- Hence where K = thermal conductivity of ice, and -£ is the temp era Cure of the aiT above the pond. Equating we get pL which on integration yields P L Lhe constant of integration being zero as z = at * = 0. Thus the time 10 obtain a given thickness is proportional to the square ot ttie thickness. Exercise.— The thickness of ice on a lake is 5 cm. and the tem- perature of the air is -2G°C. Find how long it will take for the thickness of the ice to be doubled. (For ice thermal conductivity is 0-005 cal cm- 1 sec.- 1 C 'C~\ density = 0,92 gm./cx,, latent heat -= 80 cal./i i [\Ve have *K-SH IpS-a'-* pL t= = 2 KB 10'— 5* 0-92x80 ~2 0405x20 = 27,600 sec = 7 A 40 rtt .] CONVECTION OF HEAT 29. Natural aad Forced Convection*— We have already stated in § 20 and § 24 that, there will be transfer of heat by convection in fluids unless proper precautions are taken. Convection is the transference of heat by heated matter which moves carrying its heat with it. Thus it can take place only in fluids. Free or natural convection always takes place vertically and is caused by gravity as a consequence of the change in density resulting from the rise in temperature and con- sequent expansion. ' In forced convection a steady stream of fluid is forced past the hot body by external means. Still-air cooling is ural convection ; ventilated cooling in a draught is forced convec- tion. The theoretical treatment of convection is rather complicated though the problem of forced convection is a little simpler. Never- theless convection is of great practical importance. The land and 202 CONDUCTION OF HEAT [CHAP. sea breezes, the trade winds, the tall of temperature with height in the atmosphere (discussed fully in Chap, XIT) are all examples of convection on a huge scale in nature. The ventilation of rooms and the central heating of buildings in winter arc some examples of forced convection in e very-day life. 30, Natural Convection, — 'When a heated body is cooled in air. all die three methods of heat transference are acting sirimltaneoush. But the air is a very poor conductor of heat, and radiation is import- ant only for large differences of temperature ; thus the chief means of heat loss is convection. The mechanism of this heat loss is easy to understand but the derivation of a theoretical formula is extremely difficult. The problem is therefore best studied experimentally or by a recourse to the method of dimensions. In [lie film theory of cooling by convection the mechanism of heat loss is somewhat as follows :— It is believed that the whole sur- face of the cooling body is covered by a thin layer of stagnant fluid adhering to the surface. In natural' convection this film is perma- nently present while in forced convection this is being continuously wiped off and renewed, Thus in natural convection the heat has to flow across this film of air and the amount lost will depend upon the thermal conductivity of the air and the temperature difference between the body and the air. The heat transmitted will raise the tempera- ture of the air > convection which will he opposed by via • forces. Thus the heal loss will depend upon the specific heat, expan- nt and viscosity of the air. experiments on natural convection was edit. They observed the rate of cooling oi ii :i large coj iper globe surrounded by a hath at a fixed riments were first, done with the globe eva- when the heat loss w is due to radiation alone and the rate ■■ fall of temperatui Bound to be -5-*c-^>. (31) where ' a constant depending upon the nature of the surface and Q, 8 the temperature of the thermometer and the enclosure res- pectively. Next the globe was filled with different gases at different pressures, and rate of cooling observed. Subtracting from this the radiation loss, the loss due to convection alone was computed and found to he -^^nipHd-6^*. m where p is the pressure of the gas and nij c are constants depending upon the gas, Thus the rate of loss of heat due to natural convection can he written in the form - f= k(d-o )^ VL1J. CONVECTION OF HEAT 203 where k is a constant depending upon die gas and its pressure, and 5/4 ha^been written instead of 1.233. This Js called the five-fourths power law for natural convection which can be also deduced theore- tically. For forced convection experiments show that the rate of loss of heat is proportional to die temperature excess (<?-0 o ) . It will be thus seen that Newton's law of cooling holds for forced convection even for large temperature differences but is not true for natural convection. As shown in Chap. XI it holds for radiation provided the temperature difference is small. Books Recommended 1. Glazebrook, A Dictionary of Applied Physics, Vol. T. 2. Ingersoll and Zobel, Mathematical Theory of Heat Conduc- tion (Ginn, 1913). 3. Carslaw and Jaeger, Conduction of Heat in Solids (10 : Clarendon Press, Oxford. CHAPTER IX HEAT ENGINES L Introduction to Thermodynamics*— Thermodvn amirs is liter- al I j the science that discusses die relation o£ heat to mechanical energy. Rut m a broad sense, in comprises die relation of heat to other forms of energy also, such as electrical and chemical energy, it energy, etc. The principles of Thermodynamics arc very gem m their scope, and have been applied widely' to problems in Physics, Chemistry and other sciences. The theory of heat engine* r| ; Q an integral part of the subject, and as the 'early developments were largely in connection with the problem of conversion of heat enemy S S eC ^ 1C W ?™' we sha . 1] ^gin the study by devoting a chapter to the Theory of Heat Engines, HEAT ENGINES* 2. The progress of civilisation has been intimately bound up with mans capacity for the development and control of power History tells us that whenever man has been able to make a great discovery leading to a substantial increase in his power, a fresh epoch m civilisation began, * he present ag sometimes been styled as the 'Steam Aee' 1 influence exerted by the invention of the ^ of human proves, Tn this chapter. u make a brief survey of this "great event." present time we know that Heat is a kind of motion r motion disappears it reappear, as heat, and experiments hat 1 calorie of heat is equivalent to 4.18 x 10* ergs of work. e question naturally arises : "Can we not reverse the process? Can we not by some contrivance, convert heat which is in so much excess ;»t m, to useful work?" This is in fact the function of heatengines to wo r r arE g ° ing t0 They "* contrivailces to convert heat "N»m£ J? wJHSft to «*» to » u jw« Profoundly ignorant of the irfrhlffnl 4-t he T thC Pr0 ? em did not F* 6 * *** to them ! r t,^ Z hey \ bwever ' ? bseiTed that B«wDy when bodies verTbodv C ?' P P ° Wer " D ° ay take three exa Wfe familiar t™ ,!; When WaL T i 5 ^ iled in a dosed ketdc > rhe lid is blown off place to acknowledge our grateful thanks to the author. TX.j EARLY HISTORY OF THE STEAM ENGINE 205 Z When gunpowder or any explosive is exploded a sudden im- pulse is created which may be utilised for throwing stones, cannon balls and for breaking rock. 3. High velocity wind can be made to do work, e.g., from early times sails have been used for the propulsion of ships, lor driving mills (wind-mills) . We know that such winds arc due to intensive heating of parts of the earth's surface by the sun. The three illustrations chosen above have served as the starting point for three different classes of engines which convert heat to work, viz : (I) the steam engine widely used for locomotion and in in- dustry, (2) the internal combustion engines used in motor cars, aero- planes, and for numerous other purposes, (8) the windmills and steam and gas turbines.* Many of the principles utilised in these engines art mute common to all classes, and we shall begin by describing the evolution of the steam, engine. Though the mechanical details" are outside the scope of this book, an elementary discussion is included for the sake of completeness and continuity of treatment. 3- Early History of the Steam Engine. — The earliest record of human attempt to make a heat engine is found in the writings Hero of Alexandria, a member of the famous Alexandrian school of philosophers (BOO B.C.— 100 A.D.) which included such famous men of science of antiquity as Ptolemy (astronomer) , Euclid (geometer) and Eratosthenes (geographer) . Hero describes a scientific toy in which air was heated in a closed box and allowed to expand through a pipe into a vessel below containing water. The water was thus forced up through another pipe into a vertical column producing an artificial fountain. There was, however, no suggestion to employ it on a large scale. In 1606, about two millenia after Hero, Marquess Delia Porta, founder of the Neapolitan Academy and one of the pioneers of scientific research in Europe, employed steam in place of air in Hero's experiment in order to produce a fountain. He also suggested that in order to fill up the vessel with water, it may be connected by a pipe to a water reservoir below. If the vessel filled uo with steam be now cooled with water from the outside, steam inside will condense, a vacuum will be produced, and water will be forced up from the reservoir, replenishing the vessel again. This principle was utilised by Thomas Savery in 1698 to construct a water-pump in g machine which is described below. He was the first man to produce a commercially successful steam engine which was extensively used for pumping water out of mines, and supplying water from wells. The principle utilised in Lhis engine is illustrated in Fig, 1 (p. 206) . V is a steam boiler. A, B and" C are valves, The operation takes place in two stages * — ♦Recently during the second world war a new type of engine based on the rocket principle was developed in Germany and Italy. In these a high velocity jet of air escapes at the rear of the machine which on account of the reaction thus produced moves forward with tremendous velocity, T 206 HEAT ENGINiS j ( ■. i \r ix.l J AM liS W> Fig. 1. -Principle cf Savory's (a) B is kept, closed and A. C are open, Steam passes from V to P and forces tlie water Eg (b) A and G are closed, and B opened. Cold water is sprink- led on P, This condenses steam in P, a vacuum is created and water is sucked up from the pit E to P, After this the operation (a) may be again performed and a fresh cycle begun, Savery*s engine could not suck water through, more than 34 Irt'i,, but it could force up the water to any height. In fact, be sometimes forced up water to a height of 300 feet. This means. that he used high pressure steam up to 10 atmospheres. This was a risky procedure though Papin had shown about 1680, how the risk in using high pressure steam could be minimised by the introduction of the safety valve. Papin, a French settler in England, had discovered a method of softening bones by boiling them in a closed vessel under pressure. as we know, raises the boiling point of water to about 150°C and makes the watei a very powerful solvent, Papin invented the boiler to re vent his vessel from being blown up by high pressure steam. This is shown in Fig. 2. valve consisted of a rod LM ted at I. and carrying a weight N at the nd. Tt pressed down the valve P which exactly fitted the top of the tube HH leading from the inside of the boiler. Whenever the mii pressure exceeded a certain limit, it forced up the valve P and the excess steam would rush out. By adjusting the weight of or its distance from L, the maximum steam isure could he regulated at will. 4. Newcomen'3 Atmospheric Engine* — The next forward step was the invention of Newco- men's Atmospheric Engine which was designed to pump out water from mines and wells, and was in practical use for more than fifty years. This ie is interesting from the historical point of view since it directly led to the grear inventions of James Watt, and it employed for the first time, the cylinder and the piston, which has been a feature ol steam engines ever since. Fig. 3 illustrates the Newcomen Engine. Fig. 2.- iPap&i'g Safety Valve. A is the cylinder, T is the piston suspended by a chain from the lever pivoted to masonry works. The other arm of the lever carries the piston rod W of the water pump which goes into die well. There is a counter-weight M to balance the weight of the piston T. The problem is to move the piston T up and down. This was achieved as follows : — Starting with the piston T at the bottom of A, steam is introduced from the boiler B which forces the piston up till it reaches die top. The steam is shut off by the tap 1), and cold water sprayed through F which condenses the steam in the cylinder. Vacuum is produced inside the cylinder and consequ- ently the atmospheric pres- ide forces down the piston, I J is again opened and a fresh cycle begins. The water in the cylinder A drains out through a side pipe. For closing and open ing the valves automatically, a allel motion guide was provided which carried Fig-. 3. — Nciwcomeir'.-, ALii:-:.ii.u-. lie Engine. mechanism lor automatically operating the valves. The story goes that the invention was due to a lazy boy who was employed to close and open the valve by hand, but who tied a parallel rod to the swing- arm of the lever, and connected it bv means of cords to the valves, and leaving this rod to do his work enjoyed himself all the while in playing. Whatever may be the origin, the parallel guide has been a permanent feature of steam engines ever since. In the Newcomen Engine, the useful work is done by the atmos- pheric pressure while steam is only employed to produce vacuum, hence the name atmospheric engine. It is easily seen that it is very wasteful of fuel. 5. James Watt. — James Watt is commonly credited with the- discovery of the steam engine. The circumstances which directed his attention to steam engine are pretty well known. He was an ingenious scientific instrument maker at Glasgow, and in 17G.-5 he asked by the professor of Physics at the Glasgow University to : I .:• a mien Engine belonging to them which had never worked well. Willie i : : i d in the' repair of this machine, the idea occurred to him that the Newcomen Engine was awfully wasteful of fuel, and being of an inventive temperament, he began to ponder and experi- ment on the production of a better type of machine, He was thus f ENGINES CHAP- led to a series of investigations and contrivances which gave the steam engine its present form and rendered it a mighty factor in the onward march of industry and civilisation. We are describing some o! inventions below. 6. Use of a Separate Condenser,— Watt observed that a large part of the expansive power of steam is lost on account of the fact that the cylinder is alternately heated and cooled. The expansive power of steam depends upon its temperature. Now when the steam enters the cylinder, which has been previously cooled to create a vacuum, sonic heat is taken up by the cylinder in becoming heated and is not converted into useful work. The temperature ot steam falls and its expansive power is diminished. Another disadvantage in using the cylinder as condenser is that cold water entering the cylinder becomes heated and exerts appreciable vapour pressure. thus preventing the formation of a goad vacuum. The "problem was to cbndensi the steam without, cooling the cylinder. Watt achieved this by the use of a separate condenser. The principle of the separate conden- ser is illustrated in Fig. 4. A A is the cylinder in which the piston P moves -to and fro. The piston is provided with a ; i n he PQ carrying a. valve Q at the nser - end such that Q allow* steam to go out ,ni is closed by the atmospheric pressure when there is vacuum in- Eng with the piston P at the bottom of the cylinder, R r j are and the part of the cylinder above P is filled with nit the air and residual steam through Q. Then S id T opened. The steam ■ Lwn into the condenser C which had been slv evacuated by the pump B f and is there ilensed by the cold water surrounding the condenser. Come- ly a vacuum is produced above P and steam from below pushes die piston P upwards, doing work on the weight. W- Then T is closed. S opened and P is drawn down to the bottom by W and the cycle begins afresh. The pump D serves to remove the air and water produced from steam in G. To keep the cylinder hot, Watt further surrounded the cylinders by a steam box and wood. Now-a-days the cylinders are jacketted tth asbestos or some badly conducting substance, and then covered with thin metal sheets. 7. The Double-acting Engine.— In the Newcomen engine we have seen that the atmosphere pushes down the piston. Shortly afterwards Watt employed steam instead of the atmosphere to pull the piston down. The raising of the piston in the subsequent stroke was brought about by a counter-weight attached to the other arm of To face p. 208 Watt (p, 207) James Watt, bora in Scotland in 1736, died in 1819. His important work is the masterly perfection of the steam engine which Increased the powers of man ten times and ushered the 'Industrial Revolution." IX.] t.i-lLIo.VlIOX OF EXPANSIVE POWER OF STEAM m Cabnot (p. 213) Nicolas Leonard Sadi Carnot, bom on June 1, 1796 in Paris, died of cholera on August 24, 1832, He introduced the conception of cycle, of operations for heat engines and proved that the efficiency of a reversible engine n maximum the beam. For these operations lo be possible the upper end ol I cylinder must be closed. Watt achieved tiw ant o a steam- efat stuffing box which is full of oily tow. This is kept Ughtly press- ed 3 against trie piston so that the piston can move through the cover without loss am. This was the so-called smgk-actmg Wat i Watt, however, soon realised that in this engine no done'hv steam dining that stroke in which the piston was raised up by the "action of the counter-weight. He saw that the power could be a mately doubled if during this useless stroke* steam is aa- mitted to the lower side and the upper side is connected to weefflj- i ;er. This is achieved in the double-acting engine, invented by Watt, with the aid of a number of valves, A modern double-acting cylinder . is shown in Fig, 5. The cylinder has ports or holes \, B, near its each end and between these lies another port E leading the exhaust or condenser. To the cylinder is fastened the steam chest C containing the It-slide valve S. Steam from the boiler enters the steam chest at the top. In the position (a) steam enters the cylinder through the port B and pushes the piston to the left, thereby driving the steam in front through A to the exhaust *,. AS the piston moves to the left the slide valve moves to the right and . 5.— Double-acting cylinder with slide valve. closes both the ports A and B For a time, and later when the i v.-* the extreme left position, B is closed and A opened. _ Steam then enters through A forcing the piston backward and > ng the The double- kinds of steam then enters through A forcing the p steam in front to the exhaust. This is shown at (b) , acting engine is now universally employed in all engines. ,-,.,-, r The timely action of the slide valve is adjusted by means of an entric wheel attached to the moving shaft (see Fig. 7). powerful engines as in locomotives the slide valve is often repla< by a piston valve which is very similar. 8. Utilisation of the Expansive Power of Steam.— Watt's another at inventi .the so-called expansive working of steam. saw that if steam is allowed to enter the cylinder all the time n is movi • tfards, the steam pressure in the cylinder wih „e the sac i the boiler and though we get a powerful stroke, the expansive power of steam is not utilised If, however, the steam 14 210 HEAT ENGINES CHAP. is cut off when the piston has moved some distance, the piston would complete its journey bv the expansive power of steam, whose pressure will in consequence be reduced to almost that of the condenser. Thus more work is' obtained from the same amount of steam by allowing the steam to expand adiabatically and hence the running of the machine becomes considerably economical. It is thus of great advantage to use high pressure steam. It was mentioned in the last section that the slide valve closes bo lit the ports A and B when the piston has moved some distance. From this instant to the end of the stroke the steam is allowed to expand adiabatically. 9. The Governor and the Throttle Valve.— Another simple but very useful invention of Watt was that of the governor. This is a piece of self-acting machinery which controls the supply of steam from the boiler into . . cylinder, and ensures smooth running of the engine at a constant spe Watt's governor is shown in Fig. 6. S is a vertical spindle which is made to revolve by means of gearing from the en- gine shafts. It* speed, there- fore, rises or falls with the engine speed. It carries a pair of heavy balls which are fasten- ed to S by rods pivoted at P. The balls rise on account of centrifugal force as the spindle rotates, and as they do so they pall down a collar C which slides smoothly in the spindle S. The ne end of a lever L, pivoted at Q. The other end p in the steam ipe called the throttle valve. As ulled down, thi ■ Live tends to close the tap, the steam a] md the engine speed falls. If the engine speed is too diminished, the balls fall down, C is pushed up, and the throttle valv. tting more steam, and the speed goes up. Thus the governor automatically regulates the speed at which the engine runs. Improved forms of governors are now employed in engines. 10. The Crank and the Flywheel— Watt was the first to convert the to-and-fro motion of the piston into circular motion by means o£ the connecting rod and the crank. Thus the steam engine can he made to turn wheels in mills, work lathes and drive all kinds of machinery in which a rotary motion is needed. The connecting-rod R and the crank C are shown in Fig. 7 at (a). The crank is a short arm between the connecting-rod ant! the shaft S. The connecting-rod is attached to the piston rod consequently takes up the to and-fro motion of the latter. As ing-rod forward it pushes the crank and thereby rotates the shaft S. In the return stroke the circular motion is. (Ill-: t IX.] MODERN STEAM ENGINES 211 I'iij. 7. — Crank, Eccentric and* Flywheel. completed. There are., however, two points in each revolution when the connecting-rod and the crank are in the same line and the pis- ton exerts no turning moment. These are called the 'dead centres/ At two points when the crank is at right angles to the connecti ng-rod the torque is maximum. To prevent the large variation in the magni- of the torque producing variations in the speed of the shaft during a single revolu- tion a big flywheel F, shown at (c) s is attached to the shaft. The flywheel on account, of its large moment of inertia carries the crank shaft across the dead centres ', in fact, it ab- sorbs the excess of energ) sup- plied during a part of the half- revolution and yields had I same in the remaining part of the half-revolution when less energy is supplied. Thus the flywheel acts as a reservoir of energy which checks variations during a single stroke, while the governor prevents variations from stroke to stroke. Another mechanism to convert the to-and-fro motion into cir- cular motion or vice versa is the eccentric, shown at (b) , Fig. 7. It consists of a disc mounted off its centre on the shaft S and sur- rounded by a smoothly fitting collar to which the rod is attached. The behaviour is as if there was a crank of length SC. . Such an eccentric is mounted on the shaft carrying the flywheel (shown at c) and works the slide valve. The effects can be properly timed by suitably mounting the eccentric on the shaft. The essential parts of a simple engine are shown in Fig. 8. They will be easily followed from the figure. 11. Modern Steam Engines. — Since the time of Watt many important innovations have been introduced into the steam engine though the main features remain the same. The innovations were needed in order to suit the circumstances of ever- widening applica- tion of steam engines to various purposes. Watt always used steam engines with low steam pressure., and of a static type. He was evidently afraid of explosions. But engines using low pressure steam are comparatively inefficient, as we shall see presently, and in modern times high pressure engines have almost replaced the old Watt engines necessitating the construction of special type of boilers. Condensers in modern engines consist of a number of tubes containing cold water kept in circulation by means of a pump, and JT 212 HEAT ENGINES CHAP. Slide Valve Crank Piston Piston Rod Gudgeon Pin Wheel are further provided with a pump to remove the air and water pro- & bv steam on condensation. m Again in powerful engines the Steam Ch©sT_ high pressure steam is not allowed to expand completely in a single cylinder. The steam is partly ex- panded in one cylinder and passed on to one or more cylinders where the expansion is completed. Such engines are known as compound engines and may consist of three or four cylinders. Richard Trevithick was. the first to construct a 'locomotive 1 , i.e., a steam engine which con Id draw carriages on rails. He could however, push his inventions to financial success. It was left to George and Robert Stephenson, father and son, to construct the First successful locomotive — the "Rocket", and run the first rail- way train in 1829, between Liver- and Manchester. Robert on was the first to apply the steam ei Q ships in 181;:. 12. Efficiency of Engines and Indicator Diagrams. — The earlier ventors of steam engine had no clear idea of the Nature of Heat, ra rather than physicists, they did not make seri- -. attempts at understanding the physi involved in the running of a steam engine. They measured efficiency by finding out the quantity of coal which had to be burnt per unit of time in order to develop a certain power. Tin's was rather a commercial way of measuring efficiei An absolute measurement of efficiency is obtained from the first law of thermodynamics. A heat engine is merely an apparatus for rsion of heat to work. THe heat supplied is obtained by finding out the calorific value of the fuel consumed by burning a sample of the fuel in a bomb calorimeter (p. 63). If Q be the calori- fic value of the fuel consumed per unit of time and W the power developed, we can define the economic efficiency t; as the ratio be- tween Q and W, viz. rj = W/JQ. Accordingly tj is the fraction of the heat, converted to work. For an ideal engine, ■>} should be unity. But actual experience shows that i) is rather a small fraction. In Watt's days, it was only h% ', now-a-days even in the best type of steam-engines, it hardly exceeds 17%. The question arises whether this lack of perfectness is to be ascribed to the bad designing of heat engines, or whether there is Shaft £ Eccentric .—Alain parts i ;ine. THE CARNOT ENGINE 213 ix.l something in the very nature of things which prevents us from con- verting the whole amount of heat to work. This question was pondered over by Sadi Carnot about a hur> m -,,i years He showed that even with an ideal engine, it is ■possible to convert more than a certain percentage of heat to work. It is very convenient to represent the behaviour of an engine by an indicator diagram and hence iu discussing the theory and per- formance of heat engines this is always done. Suppose a certain amount of gas is contained in a vessel at a certain perature and pressure and occupies a certain volume. Evidently the state of the substance is uniquely represent- 1> ed by assigning its pressure and volume. Thus we can represent the state of the gas by a point A (Fig. 9) on a graph such that the abscissa of the point re- presents the volume of the gas and the i :ate represents the pressure. Let the pressure and the volume of the gas be changed to that corres- Ffe. 9. Ti: ponding to the point B and suppose the pressure and the- volume i rnout this change are represented by points on the line All. Then this operation is represented by the line AB on this dur Such p-v diagrams arc known as indicator diagrams* \s proved on p, 4fi the work done by the gas in expanding against a pressure p is p*S. In this case since p changes from point to point ral work done by the gas in expanding from :•, to v* is equal to f v * pdv and is evidently equal to the area AabB, The work is taken to be positive if the diagram is traced in the clockwise direction. It is thus clear that the indicator diagram directly indicates the work done bv the engine during each cycle of operations, the work being equal to the area a by the indicator diagram, indicator diagrams are therefore of great use in engineering practice. The indicator diagram gives the work performed by the piston per stroke. Multiplying it by the number of strokes per second i.e. bv twice the number of revolutions of the shaft, we get the power indicated by the indicator which is generally expressed in horse-power _;,.,. p a The power actually delivered by the engine is measured by a brake dynamometer and is called the brake horse- wer, the ratio of this to the _ indicated horse-power is called the medio:; in; I efficiency of the engine. 13. The Carnot Engine.— -As we have seen, the function of the steam engine is to convert the chemical energy stored in coal to energy of motion by utilising the expansive power of steam. 214 HEAT ENGINES [CHAP. The machinery necessary for this purpose is, however, so com- plicated that one is apt to lose sight of the essential physical prin- ciples in the details of mechanical construction. Lee us, therefore, discuss the physical principles involved in the running of a heat engine. Three things are apparently necessary, viz., a source of heat, a working substance, and machines. In the steam engine, the source of heat is the furnace where heat is supplied by the burning of coal. But we may get heat by a variety of other means, e.g. s by burning oil, wood, naphtha, or even directly from the sun (solar engines) , or from the inside of volcanoes (as is sometimes done in Italy). We can, therefore, replace the furnace by the general term "reservoir of heat/* For steam, we can use 'the general term 1 working substance," for any substance which expands on heating can be used for driving heat engines. As a matter of fact, we have got hot air engines in which air is heated by a gas burner or a kerosene lamp, and pushes the piston up and down as steam would do. In addition to the three requisites mentioned above, we require a fourth onc,_ viz., the possibility of having a temperature difference. 1 his at first^ is not so apparent, but can easily be made clear. In a hot air engine, the heated air can push the piston outward since the air outside is at a lower temperature. If there were no difference of temperature, no difference of pressure could be created, hence the machine would not work. We, therefore, requhe not only a sou of heat, but also a sink, i.e., a heat, reservoir at a lower temperature In .steam engines the surrounding air acts as the sink of heat or con i bserved that the fund the machine is to extract a certain quantity of heat Q horn the heat reservoir F, convert a part of it to work and transfer the rest to the heat sink G. He also showed how these operations should be carried out so that the efficiency may be maximum. Since whenever there is a differ- ence of temperature, there is a pos- sibility of converting heat to work, the converse is also true, i.e. if we allow heat to pass from F to G by conduction, wc miss our opportunity' of getting work. Hence we must extract heat from F in such a way that loss of heat by conduction is reduced to a minimum. Carnot, m« m ti - a 1 ^ lefore, thought of the following s ' 10 :~V^. ld f ' £ arnot en «'» e ideal arrangement, with indicator diagram. t 5 ^ rr,. ,,-s, * F T ' s a Jieat reservoir at tempera- ture T (Fig. 10) , G a heat-sink at T f , S is the cylinder of the engine containing a perfect gas instead of steam as the working substance and fitted with a non-conducting piston. The walls 'of the cylinder EL] THE CARNOT ENGINE 215 are impervious to heat but the bottom is perfectly conducting. The behaviour of the working gas is shown by the indicator diagram showing the pressure and the volume of the gas at any instant. i>et the following steps be performed :— (1) Let the initial temperature of the gas within S be T and let it be pitted in contact with F, and die piston moved forward slowly Ah the piste, : - the temperature tends to fall, and heat will pa^ from F to S. The operation is performed very slowly, so that the temperature ol the gas is always 7\ The. representative point on the indicator diagram moves from A to B along an isothermal curve. 1 he heat O cted in this process is equal to the work done by the piston in free expansion, and is given by = = f B pdv = RT log, £ - area AabB. . . (1) (2) F is then removed and H, which is simply a non-conducting cap, is applied to the cylinder, and the piston allowed to move for- d (by inertia). Then the *as will describe the atflabatic BC and will faU ill temperature. We stop at C when the temperature has fallen to P. 'I he work done by the gas is given by where piP — K"=£ & o v = AbV< Since the pressure is now very much diminished, die gas has lost its expansive power, hence in order to enable it to recover its capacity Tor doing work it must be brought back to its original con- dition. To effect this we compress the gas m two stages: first, isothermalh all ng the path CD, and then adiabaticaUy along DA. The point D is obtained by drawing the isothermal T through U and the adiabatic through A. (B) During the isothermal compression, the cylinder is placed in contact with the sink G at !P. The heat which i s developed owing to compression will now pass to the sink. This is equal to the work done on the gas and is equal to = area CcdD. . (3) -. 0' = \°pd»=* tfT'log,-^ (4) The cylinder S is now placed in contact with II and the gas is compressed adiahatically. The work done on the gas by adia- batic compression is Wi = f "/wfc = ~( T- T') - area DdaA. . It is thus seen that m z t=W4, m 210 HEAT ENGINES The net work done by the engine W — w x -\. w s w 4 — area A1JCD, [CHAP. (6) The last result can be written down directly from the first law of thermodynamics. Since B and C Ik on the same ad i aba tic, we have by equation 24, p. 48., 1 ■:'...-■" ■'">■.- - 1 (£P" <?> — I ■/-• J — Pi the adiabatic expansion ratio. or Simi • n prf or = :r^= r .» the isothermal expansion ratio have, therefore, Q = RT log r, £' = i2:r log r and Tr-0-/>' = i;.;;r-T') log • r. . r 3"' ""■•/-: Hence -afV'j (S) (9) of the Cartiot en: W T' -HIP < 12) analysed the moi ion of the Camot cycle we now I to .show that (I) it is reversible at cadi stage. (2) that no engine can be more efficient than the Carnot engine, m that the re of the working substance is immaterial. 14. Reversible and Irreversible Processes,— A process is one which can be retraced in the opposite direction to that h substance passes through exactly the same states in all t8 as in the direct process. Further, the thermal and mechanical effects at each stage should be exactly reversed, U., the amounts of heat ived and of work done in each step should be the same as in the direct process but with opposite sign. That is, where heat is absorbed in the direct process it should be given out, in the reverse cess and ; : -, th and where work is done bv the working sub- stance m die direct process, an equal amount of work should be DC] RSIBLE AND IRREVERSIBLE PROCESSES 21? done on the working substance in the reverse process. Processes in which this does not take place are called irreversible. For clarity we may add some examples of reversj F 1 '■■••-■'- • - transfer of heat from one bodv to another can be reversible only when the two bodies are at the same temperature. In case of two bodies at different temperatures, the transfer of heat occurring by conduction or radiation cannot be reversed and the process is irn ible. We shall now consider examples of reversible processes. It is r from the above definition that the process of bringing an elastic substance into a definite state of stress very slowly is revers because for a given strain the substance has always a' definite stc A convenient mechanical example of a reversible pro* afforded by the performance of a spring balance in the following way; When the spring is very slowly stretched work is done upon it. If, on the other hand, it is allowed to contract slowly bv the same amount the same amount of work is done by the spring, for die work done in increasing the length of the spring by Si is equal to the product of the force F and U. Both these quantities depend upon the si of the spring at the instant and ave the same value whet] l f c s P n ' : : ding or contracting. Thus the work done upon n f.. be equal to the work done bv the ■ wlu - the reverse process and the process [ f n ■'■• ■■'< however, that the stretching must be d or reduced gradually by the c 'ion of a- force • mftnitesimaUy at every instant, from the stn developed m the spri ,, part of the work will be spent in setting up vibrations of the spring and this will ability Such a process is called a quasi-static process and consists essentially of a ,. m of equilibrium states. The case of elastic fluids is analogous to that of the spring To- :v volume of the fluid there corresponds a definite stress pressure, so that the amounts of work done during a balanced expj sion or compression are equal. This is an important example o£ a reversible process. It is important to note that the expansion should be balanced otherwise whirls and eddies may he set up in the fluid which will gradually subside on account of flui< i :i ion with the pro- duction of heal and thus a part of the mechanical work would be lost. Such expansion or contraction may be either isothermal or acLxababc and can be brought about easily bv applying pressure on piston enclosing the fluid and adjusting the pressure to differ from the fluid pressure by an infinitesimal amount. Examples of irreversible processes are (I) sudden unbalanced expansion of a gas, either isothermal or adiabatic, (2) TouIcThotri- expansion, (3) heat produced by friction, (4) heat generated when a i ent flows through an electrical resistance, (5) 'exchange of heat 5? 1 "52! ! ? odles ,^ differ e nt temperatures by conduction or radiation (n) diffusion of liquids or gases etc. Examples (1) and (2) exhibit internal or external mechanical irreversibility, (4) and (5)' exhibit thermal irreversibility, and (6) exhibits chemical irreversibility. 21 a HEAT ENCINF5 [CHAP- A reversible process may be represented by a line on the indicator diagram (p, v) out an irreversible transformation cannot be so represented. 14. Reversibility of the Carnot Cycle*— It is now important to notice that the Carnot cycle is reversible at each stage, i.e., instead of abstracting the heat Q from a source T and transferring a part Q' i sink T' and converting the balance Q - Q' to work, we can proceed in such a way that the machine abstracts the heat <T from the sink at T, then we perform the work W on the machine, and Q is trans- ferred to I source at 7\ This is done by proceeding along the reverse route ADCBA, i.e., first allowing the gas to expand adiabati- cally from A to D, then allowing it to expand isothermally from D to C in contact with the sink at V, the heat (T being extracted in the process. Then we compress the gas adiabatically till we reach the point B (temperature T) . Next S is placed in contact with the source, the gas is further compressed isothermally till we reach A, and heat (Ms transferred to the body at the higher temperature T. The machine, therefore, acts as a refrigerator, :>,, by performing the work W on it, we are depriving a colder body T* of the heat O . An engine in which the working substance performs a reversible cycle is called a reversible engine. Engines in which the cycle is versible are called irreversible engines. Fur the Carnot cycle to be reversible it is essential that the working substance should not differ sei erature from that of the hot body and the condensi l it is exchanging heat with them. There should be of heat by conduction in the usual sense, for it. would be [b] . This ires that the isothermal processes AR, CD bed indefinitely slowly, and the source and the con- arge capacity for heat so as not to change in temperature during the proot ain the piston should move very slowly without friction. The changes in volume should be brought about by very small changes in the load on the piston in both the .isothermal and adiabatic processes, so that the difference between the external pressure and the pressure of the gas should always be infinitesimally small. Tims the Carnot cycle postulates the existence of stationary states of equilibrium while in an actual process the physical state is always changing. Further there should be no loss of heat by conduction from the to the piston and cylinder. Tt will thus be seen that the Carnot cycle with its perfect reversibility is only ideal and cannot be realised in practice, Nevertheless, it should be noted that for theoretical purposes die deviations from the ideal state may be lightly neglected if they are not an essential feature of the process and if they can be diminished much as may be desired by suitable devices and then corrected for. 16, Carnofs Theorem.— The idea of reversibility is of the great- est importance in thermodynamics for the reason that, working between the same initial and final temperatures, no engine can I IX.] CARNOT S THEOREM 219 ■ent than a reversible engine. This is known as Garnet's theorem. We now proceed to prove this important theorem, Suppose we have two engines R and S, of which R is a reversible engine and S an irreversible engine. If possible, let S be n efficient than "R, Suppose S absorbs the heat £) from A, converts l W f to work and returns the rest, viz., Q~W to the condenser. Let S be coupled to R (Fig. 11) and be used to drive R backwards. We are using R as a refrigerator. It thus abstracts a certain amount of heat from B, has the work W performed on it, and returns the same heat Q to the source A. The amount "of heat that R abstracts from B must equal Q-W* Now since S is assumed to be more efficient than R, W>W and hence Q— FF> Q - W r , vi&, R abstracts more heat Iron: B than S restores to it. Thus the net result is that the compound engine RS abstracts heat (W-W) per cycle from E and converts the whole of it to work,f while the source is un- N /* Hot Bod 1 ? A > v R ^ ill •-.-., y B Q-W' \ • II,— Coupling a reversible and an irreversible engine. affected. We are thus enabled, by a set of machines, to deprive a body continuously of its heat-content and convert the whole of it work without producing any change in other bodies. The machine would tii us work simultaneously as a motor and a refrigerator and would be the most advantageous in die world. It does not violate the first law for we are creating energy out of heat. Impossibility of perpetual motion of the second type. But still the process is quite as good as perpetual motion! of "the first kind. for heat, is available to us in unlimited amount in' the atmosphere, in the soil or the ocean, and if the process were feasible, it would give us all essential advantages of a perpetual motion machine, viz., that of getting work without any expenditure, though not without enere Human experience forbids us to accept such a conclusion^ Hence we conclude that no engine can be more efficient than a reversible engine. Again, if we assume that a reversible engine using a particular * Reproduced from Ewing's Steam Engine by the kind permission of Messrs Maemil3a.ni & Co. t X!l'- £ work ^' ~ ^ which is available, mav he used in driving a motor. J this was a term in use amongst the medieval philosophers who. thought that a machine might he invented some day which will create work out of nothing for some mechanical contrivances which , v c-re intended to produce perpetual motion see Aml-ade, fiuw, 3 «gc 14. The gradual evolution of the Law of ration of Energy lead; : to the first law of thermodynamics showed that is purely chimerical, since energy can never be created out of nothing ;'," V 11 -''' ^ transformed from one form to the other. 1 1 Tins result of human experience is the fundamental basis of the Second Law of Thermodynamics. In fact, Kelvin stated the second Jaw in this form t*or fuller discussion, set Chap. X. 220 HEAT liNGINE* [CHAP, working substance is more efficient than another working with a differ- ,1,1 substance, we arrive by a similar argument at the aame absurd result. Hence the efficiency is the same for all reversible engines this is the highest limit ; efficiency of any engine that can be constructed or imagined. This is Camot's theorem. Thus rever- sibility is the criterion of perfection in a heat-engine. Hence we sec that the efficiency of a reversible engine is maximum and is lnde pendent of the nature of the ing substance. We chose perfect as our working substance, since its equation of state being known, we can easily evaluate 17. Note : — It may be mentioned that Carnot was ignorant of the true nature of heat at the time when he published his theorem of efficiency- He followed the old caloric theory in his speculations according to which the quantity of caloric O contained in a gubsta was invariable. It was Clapeyron who showed that Camot's argu- ments and result* remained intact when the Kinetic Theory of Heat was introduced. 17, Rankice's Cycle. — In the steam engine, as we saw in the foregoing pages, the working substance is a mixture of water arid water vapour. ° If, 'however, we perform the same Carnot cycle with this fluid as with perfect gas the efficiency would also be the same {Camot's theorem) . But the Carnot cycle was performed with the working sub- stance always in the same cylinder. We have already seen that with a xi vent, the je of heat consequent on alternate heal and of the cylinder the modem engine-; are provided v. Still, howevi work- ice were to perfon e the efficiency wouldbe with the organs so separated the adiabatic corny working substance in the last stride of the cycle becomes imp] : e . Hence the cycle is modified into what is known as - cycle. The Rankine cycle is represented in Fig. 12. A'B represents the conversion of water into steam in the boiler at temperature F 3 and e pi and its admission into the cylinder, EC the adiabatic "on in the cylinder, CD the transfer of steam from the cylinder t:r at T 3 and pa and its condensation, and DA the transfer by a separate feed-pump to the boiler. This separate feed- pump transfers the condensed water at D at T 2 and p s to the boiler and the pressure is consequently raised From p% to p x . At A the water is heated from T 2 to T 2 in the boiler and the cycle begins afresh. The indicator diagram for an engine per- forming the ideal Rankine cyi repre- sented by ABCD and this also represents the work done by the engine. The area Fig. 1 -The Rankine cycle. FADE lily called the Jecd-puinp term, is extremely small and is generally lected. The work done by the engine may, therefore, " be put equal IX,] TOTAL HEAT OF STEAM 221 to the area. FBCE. This area can be calculated in much the same- way as on p. 215 if we know the equation of state of steam. This is cumbrous and in engineering practice a simple procedure is adopted. Let us introduce the total heat function H = U ■{• pV. Then dll = dU + pdV -I- Vdp — Vdp for an adiabatic process, since in this case (Chap. II, §23) \.pdv=a. Integrating between the limits B and C (Fig. 12) we get ight-hand side represents the area FBCE. Hence the work done in Rankine's cycle per gram-molecule of steam is approximately equal to the heat-drop H n —He- This will hold whether the steam is super- heated, saturated or wet. The heat taken in by the working substance is that required to convert water at p y and T z into steam at p t and TV This is equal to H B - [H D | (/>i • - p^ V w \-H n II n approx. where 11%, Hn denote the total heat of steam at pi, T } and of water at y. , -lively, V w the volume of v*»ter at D. The term (pi — pi) V, v 1 nts the area FADE. The efficiency is Hs — Hq The values of the total heat function are readily obtained from steam tables or charts which have been prepared for engineering work as explained in § IS. It will be v seen that the efficiency in Rankine's cycle is less than in Carnot cycle for in the former some heat (viz., that required to heat the feed-water in the boiler from T 2 to T 7 , is taken at a lower tem- perature. In actual steam engines the efficiency is only about 60 — 70% of the Rankine ideal and their indicator diagram resembles Fig. 12 with the corners rounded off. 18, Total heat of steam. — For calculating numerical values* it is usual to assume that the total heat or enthalpy of saturated water . water in equilibrium, with i' : ; saturated vapour) at 0*C is ;••: . and calculate the changes in total heat. Then the total heat of steam at (9°C is defined as the amount of heat required to convert 1 gram of water at 0°C into steam at &°C Regnault gave for the total heat of steam H the following empirical formula H = 606,5 + 0.305 $ where denotes the boiling point. The total heat may be computed as follows : Let us raise the temperature of 1 gram of water from 0°C to its * Si.::' Keesom and data on .-"• ■ .' cmd solid ■X 1936). ICeyes, Thermodynamic Properties of Steam, inch lolid phase, (Wiley, 1936), or Calletiiiar, Steam 222 HEAT F.NGTNES [CUA!\ boiling point (t°C) , Denoting the average specific heat at constani pressure by c, the heat required for this process will be U if we neglect* the small work done due to pressure changes. Thus the total of water at t°C is H. Now let us evaporate this liquid at its boiling point t°. Consider the state of affairs when q gram of vapour has been produced leaving (1-q) gram yet to be evaporated. Then q is called the dryness fraction. The volume of. vapour produced is qv and the increase in volume is q(v-iv) where v is the specific- volume of vapour at t°C and w the specific volume of water. Hence the external work done is pq (v-w) in mechanical units. Therefore the gain in total heat during evaporation is qL— qi -\- pq (v - to) where * h the increase in internal energy when unit muss of the liquid is evaporated. Hence the total heat of steam at dryness fraction q i H=ct-\- qL. For dry saturated steam (q =. I), this becomes B = St + L. If the dry saturated steam at t"C is further heated, its temperature rises and we now speak of it as superheated vapour. In the case of am the superheating is usually produced under such conditions that can assume that the superheating is carried out at constant pres- u i the temperature of the superheated vapour to be t t °C that there are (ti — i) degrees of superheat. If c/ he the mean ilir heat n\' the vapour at constant pressure, rhe total heat of super- H s ~-a{-L+c p '(t x - INTERNAL COMBUSTION ENGINESf 19. Historical Introduction.— Like other types of heat engines the this type also dates front medieval limes, Ch. Huygens. * M ore rigorously (ill — d (« + h) — <*« + Pdv -f and (T< .'t> :QS\ -v H = Tds + vdp. + Xr dH = -c dH dT )."+(X\ *--i;^fiMfr)> and we must integrate the right-hand side from 0"C (To'K) to the boiling point T and the pressure from the saturated vapour pressure ^,> at To to the saturated vapour pressure p at T, t These are called internal combustion engines because in them heat is. pro- by combustion of fuel inside the cylinder in contrast with the steam engine which may be called external combustion engine I cans h this tribe heat is produced in the boiler. The interna] combustion engine is much more efficient than the steam engine because the working substance can he heated to a much higher temperature (20OO°C). IX.] THE OTTO CYCLE %>$ the great Hutch physicist, proposed in 1680 an engine consisting oi a vertical cylinder and piston, in which the piston would be thrown upwards by the explosion ot a charge oL : gunpowder. This would I il the cylinder with hot gases which would eventually cool, and pis: Id be forced down by gravity. Each stroke would require a fresh charge of gunpowder. This engine was never used, probably owing to the difficulty oi introducing fuel alter every explosion but the idea pcrsisusd. The discovery o£ combustible gases (coal gas, producer gas...) and of mine- ral oils brought the idea within the range of practical possibilities, as the difficulty of supplying fuel promised solution. But many long: years of practical and theoretical study were necessary before such engines became commercially possible. 7t is not possible to go into the details of these early at- tempts, and it will suffice to describe and explain the action of the two types* which have survived, viz., (I) the Otto engine in which heat is absorb- ed ai S) the '•in whij li heat is bed al constant pressure. ly BO per cent of" internal combustion engines today are of the Oi to type. We shall com- pare the action of these engine'- with that of the ideal Carnot engine in which heat is absorb- al constant temperature. 20. The Otto Cycle,— The Ottof cycle we are now going Lo describe was originally pro- d by Beau de Rochas .1 ., but the practical diffi- culties were first overcome by L876. The working of iic. will be clear from I The engine consists of the cylinder and the piston, the cyl ider being provided with inlet valves for air and gas (if gas is . lor combustion) and exhaust, valves. The opening and closing ese valves are controlled by the motion of the piston. There our strokes in. a complete cycle: ♦Sometimes internal combustion engines are divided: into (1) gas engine, (2) oil and petrol engines, ft is, however, more scientific to divide them with regard I they follow and not with regard to fuel. ' Hence ' have been divided into the constant volume, constant pressure and constant raturc type, t Nikolaus Oi I8M,), born at Schlangenbad in Germany, 3s best k: ;:.-. the inventor of the four-stroke gas engine. Fig. 13. The four strokes of the Otto • ;.iue. 224 HEAT ENGINES [CHAP. (i) The Charging Stroke. — In this the inlet valves are open, and a suitable mixture of air and gas is sucked into the . the forward m i of the piston. (if) The Compression Stroke^- During this stroke all the val are closed, and the combustible mixture is compressed adiabatics to about l/5ih of its original volume by the backward motion of I e ton. The temperature of the mixture is thereby raised to about ; a At the end of the compression stroke, the mixture is fired by a series o£ sparks. (iii) The Working Stroke, -Hie piston is now thrown forward witl oree, since owing to combustion a large amount of beat developed which raises the temperature of the gas to Lboul 2000 D C and a corresponding high pressure is de\ (iv) The Scavenging or the Exhaust Stroke,— At the end of the third'. stroke, the cylinder is filled with a mixture of gases which is useless ; iher work. The exhaust valves are then opened, and die piston moves backward and forces the mixture taut. scav is complete, a fresh charge of gas and air is sucked in and a fresh cycle begins. The tli uamical behaviour is illustrated in the indicator ■diagram (Fig. 1 It should "■ inhered that air is the working substance in Otto engine, the function of gas or petrol being merely to heat the air by its combustion. EC represents the suction stroke (gas and air being sucked at atmospheric pressure) . CD represents the compression stroke. At D (pre about 5 atmospheres, temperature about G00 C C) the mixture is fired by a spark. A large amount of heat is liberated, and the representative point shifts to A (tempera- ture 2000 a C, pressure about 15 atmos- pheres), volume remaining constant, AB sents the working stroke. At B the ust valve is c and the pre; [alls to the atmospheric pressure GV t . is the scavenging stroke. The efficiency of the engine can be ea if'ed out. The amount of heat added b C a (T a —Tj) where C 9 is the : heat at constant volume. (We suppose that this quantity retains the same etween 6GQ°C, and 2000° C. which is only approximately true.) Heat rejected = C s ( T b - T r i . Hence efficiency tj=1--=t ,J' . • . (1$) Im.lt. 14 -I Otto cycle. t IX.] thk orro CYCLE 225 From the relation TV*-* = constant for an ad i aba tic (see p. 48, equation (24), we obtain T* iVt\*- 1 T d (14) where p is the adiabatic expansion ratio. Hence r.-r.np T B -T* Therefore the efficiency ='-(1) I ',>-' - (15) We obtain a similar expression for efficiency in terms of the adia- compression ratio as we get m a Carnot cycle (see equation 12, p. 216). The question arises : why not try to make the Carnot cycle a practical possibility? ' We shall now treat this question in more detail. In desiei a machine, other factors have to be taken into consideration to Lion of theoretical efficiency. The engine must be, in the first, place, quick-acting, i.e., a cycle' ought to be completed as tunckly as possible. This practically rules out the Cajnot engine as Les up heat at constant temperature, the process is therefore and the engine becomes very bulky and heavy in relation ait. In the Otto engine, the absorption of heat is almost instantaneous. But even where quickness is not the deciding factor, there are other weighty arguments against the adoption of the Larnot cycle. Pressure inside the cylinder varies during a cycle and Lie machine must be so designed that it can withstand the maximum pressure. Hence practical considerations impose a second condition t.e the maximum pressure developed inside the cylinder durjW a cycle should not be too great. Further, we must have a reasonable amount of work per cycle. Now due to the adiabatic compression in the last stage of the Carnot cycle an enormous pressure is developed. A detailed consi- deration taking numerical values shows that working between the same two temperatures, say 2040°K and MCPK the Carnot engine ops a maximum pressure of about 1000 arm., while in the Otto engine it ; « only about 21 atm„ though of course, the efficiency is reduced from 83% to about 44%, The Carnot engine also rec a large volume for the cylinder. These considerations show clearly tAat Che Carnot engine is quite impracticable. It will have to be very bulky and very stout and the power output will be extremely spall compared to its bulk. In the case of the Diesel cycle which is described in the next section the maximum pressure developed is about 35 atm and the efficiency rises to about 55%, It is for this reason that the Diesel engine is employed in cases where we want a large output of work. 15 mi HEAT ENGINES [CHAP- DIESEL CYCLE 227 hnl KbMm D J A 21. Diesel Cycle. — Diesel* was dissatisfied with the low efficiency of the Otto engine and began investigations with the idea that efficiency of the Carnot cycle may be reached by certain other contri- vances. ' He did not succeed in his attempt, but was Led to the invention o£ another engine which, for certai rposes, presents ked advantages over' the Otto engine, >w since it is easily seen that if the com- pression ratio p can be still further pushed up f ^ would substantially increase. But in the Otto ei p cannot be increased beyond a certain value (about 5) otherwise the mixture would be fired during pression before the spark passes. In the Diesel engine no gas or petrol is introduced during com- pression. The indicator diagram tor the cycle is shown in Fig. 15. In the first stage pure air is sucked in (EC in Fig.) an* I I to ah nut one seventeenth of its volume in fig.) . A valve is then opened, and oil or vapour is forced under IS.— The Diesel • Fig. 16.— Strokes in a Diesel Engine. (o) Beginning of suction stroke; air-valve open, i :'.',' i T: entitling of compression stroke; all valves closed. (c) BcRintiing of working stroke; oil valve open. (d) Working stroke in progress. Etui at End injection ; all valves closed, (j?) Beginning of scavenging stroke; exhaust valve open. ♦Rudolf Diesel (1853-1913), barn in Paris of Get man parentage was the inventor of the heavy oil engine. He was engineer at Munich. ure. The oil burns spontaneously as the temperature in the is about 1000°C. and above the ignition point of the fueL >f oil is so regulated that during combustion, as the ptsinh forward, Lhe pressure remains constant (DA). . when the temperature has reached the maximum value, e supply of oil is cut off. The piston is allowed to ', describing die adiabatic AB. At li a valve is opened, mill pressure drops to C. CE is the scavenging stroke in which the fixture is forced out, and the apparatus becomes ready ycle. cycle can be performed in a cylinder of the type shown 16. The cylinder is provided with the air inlet, the oil supply md die exhaust valves. The action of the cycle will be very clearly folio wi ' 1 1 from the figure, instead of opening a valve at B we may allow the gas to expand F where the inside pressure has fallen to atmospheric pres- .iiii', and Lhe air mixed with the burnt gas be forced out by a back FE. As this would involve a rather large volume for the , this procedure is not adopted in actual practice. all now calculate the efficiency of the Diesel engine. Let m i"i calculate the efficiency for the imaginary cycle AF6D, Heat taken = C P ( T a — T d ). Heat rejected = C p {T f -T t ). relation p = cT v! ^ 1 ' 1 (equation 25, p. 48) we have <*-«* T d T a T m -T d m ii T f -T c f a -f d (y-l)!y T f -T c --(£:) v-i (16) (17) e for the efficiency' the same formula as for the Carnot tto cycle. But in the Diesel engine the cylinder is designed m pressure of about 34 atmospheres. This determines 1 1 on ratio because p\fp t ■- p y =34. This gives p=12.6 which is a substantial improvement on the OltO engine, ted above, we have to take the cycle DABG. Heat constant volume. Hence heat rejected is equal to c v (T b -r c ). " v " ~C fi (T A -T d ) Id h wer value of about 55$>. (18) HEAT ENGINES CHAP. The Diesel engine, therefore, consumes less End than the Otto engine, bun els it has to withstand a higher pressure it must be more robust. Also the mechanical difficulties are much greater, hut they have been successfully overcoi 22. Seini-Dieset Engines.— We have seen that in Diesel engines To avoid these die Semi-Diesel engine, also called the hot-bulb <?.' has been invented, In this air is compressed to about 15 to 20 aim. instead o£ 35 atm. This air does not yet become hot enough to ignite the oil and is consequently passed through a hot bulb where it is heated by contact with the bulb to the required temperature. The hot bulb 'is simply a portion of the cylinder which is not cooled by the water jacket. Tt consequently becomes much heated and serves to heat the compressed air. Next the oil (fuel) is injected into this hot bulb where, by the combined action o£ the compressed air and the hot bulb, it ignites. For starting the machine the hot bulb must be heated by a separate flame. After a few cycles the bulb becomes hot and begins to function properly. Then the external flame is removed. The semi-Diesel engines now-a-days are generally two- stroke oil engines without automatic control valv 23. The "National" Gas Engine,— We shall now describe a typical Otto engine. Fig. 17 shows a 100 horse-power "National" gas : ne. Fig. 17.— The "National" gas engine. The engine is a stationary horizontal one having a single cylinder A A. The piston PP is attached to the crank shaft by the connecting rod CC and (its gas-tight into the cylinder AA. H is the combustion chamber having the inlet valve 1, the gas valve G and the exhaust valve E. The cylinder, the combustion chamber and the valve casings are cooled bv the water jacket WW. L is the lubricator, V, V are rx,] STEAM TURBINES 229 plugs which may be removed in order to examine the valves. M is the ignition plug where a spark is produced by the "Magneto" and serves to ignite the charge. The various valves and the magneto are worked by suitable earns mounted on a shaft driven by the crank shaft. The working of the machine is identical with that indicated in Fig, 13. The machine is started by rotating the fly-wheel so the piston commences its outward stroke. An explosive mixture is introduced and ignited and the piston is thrown forward. Then the working follows as indicated on p. 223. 24. Diesel Four-Stroke Engines.— - These engines employ a cylin- der and valves of the type shown in Fig. 16* In constructional de- sign they are not much different from the Otto engine. They are further provided with a high pressure air blast for injecting the liquid fuel. 25. The Fuel*— Some engines employ gaseous fuels the chief of which are coal gas, producer gas, coke-oven gas, blast-furnace gas, water gas and natural gas (chiefly methane) . The common liquid fuels are petrol, kerosene oil, crude oil, benzol and alcohol. Tn the Otto i tigine, if liquid fuel is used, it must be first vaporised and then s generally employ crude oils and hence their fuel ' heap. 26. Applications. — During the brief period of fifty years the internal combustion engines have been much developed and employed for a variety of purposes. The steam engine, we have seen, is very wasteful of fuel, but the internal combustion engine is much more economical and has consequently been widely employed. The Otto four-stroke engine is usually employed in motor cms and aeroplanes. Petrol vapour is used as fuel in both cases. Four- stroke Diesel engines are largely employed in driving ships. Where very large power is necessary, the two-stroke Diesels ate in common use. Diesel engine is more efficient and more economical of fuel. Still has combined a steam engine with a Diesel engine and has obtained much greater efficiency in steam locomotives (SLill-Kitson locomotives/'! STEAM TURBINES 27. The Steam Tnrblne. — We shall now give a brief description of the engines belonging to the windmill type. Windmills are set in motion by the impact of wind on the vanes of the mill, and at first sight it may appear strange why they would be classed as heat engines. But a little reflection will show that the wind itself is due to the unequal heating of different parts of the earth's surface, hence the designation is justified. Etit wind-power is rather unreliable, hence steam turbines were invented in which the motive power is supplied by the artificial wind caused by high pressure steam. There two types of turbines :_ — (1) The Impulse Turbine in which steam issuing with great velocity from properly designed nozzles strikes HEAT ENGINES CHAP, against the blades set on a turbine wheel and in passing through them has its velocity altered in direction. This gives an impulse to the Wheel which is thereby set in rotation. (2) The Reaction Turbine is somewhat similar in principle to the Barker's Mill which, as a laboratory toy, has been known to generations of students. In this the wheel is set in rotation in the opposite direction by the reaction produced by a jet of steam issuing out of the blades set in the wheel. Though the steam turbines are simple in theory, in practice great mechanical and constructional difficulties had to be encountered, but thanks largely to the efforts of De Laval, Curtis and most of all to Sir Charles Parsons who may be described as the James Watt of the steam turbine, these difficulties were successfully overcome. To-day the turbine is a prime mover of great importance and is aim exdusivcly used as the source of power for all large power land stations for the generation of electricity and propulsion of big ships. Their great advantage lies in the fact that instead of the reciprocating motion of the ordinary steam engine, a uniform rotary motion is produced by a constant torque applied directly to the shaft. They are more efficient than other engines because there is no periodic change of temperature in any of their parts and hence no heat is wasted. The full expansive power of steam can be utilised under suitable con- ditions. 28. The Theory of Steam Jet*.— The essential feature of the turbines is that, in them the heat energy of steam is first employed itself in motion inside the fixed nozzles or bladings (in reaction turbines) and this kinetic enej is utilised in doing work on the blades. The fixed nozzles are so designed as to convert most efficiently the thermal energy of steam into its kinetic energy and produce a con- stant jet. We shall, therefore, first study the formation of jet under these conditions. We shall derive an expression for the kinetic energy imparted to the steam in traversing the fixed nozzle from the region of high pressure B (Fig. 18) to the region of low pressure C. Let the pressure, volume, internal energy and velocity of steam in E per gram be denoted by p lt v lt ih and w ± and the correspond ins; i C by fa v., Us and w z . Then the gain in kinetic energy of steam per gram on traversing the nozzle is (u> 2 2 -n> x s )/2j cah The net work done on steam in this process as on p. 138 is p^> : — p$v s while the loss in internal energy is u t -u^. Thus on th nption that the process is adiabatic we have from the first law of thermo- dynamics, rjj {w^-w^) = p t Vj-; aa^Aj— k tt , . (19) i.e., the gain in kinetic energy is equal to the heat drop. Generally the initial velocity is sensibly zero and hence die kinetic energy of the issuing jet is given by the heat drop hi - h 2 . Fist. 18— The fixed IX.] THE DE LAVAL TURBINE 281 -£^~ m^E3* If we now assume that there is no loss of heat energy through friction or eddv currents in the nozzle, the heat drop At- feu equal to the heat drop from the pressure fa to p & under isen tropic condi- tions. This can be readilv obtained from steam tables and represents under most favourable condition the maximum work obtainable from the turbine. Under ideal conditions the ordinary steam engine would also yield the same amount of work, i.e. } equal to the heat drop. The neater efficiency of steam turbines is, however, mainly due to the following two causes:— (1) in the turbine diere are nowhere any periodic changes in temperature as are found in the cylinder of the steam engine ; (2) the turbine is capable of utilising low-pressure steam with full advantage. 29. The De Laval Turbine.— The velocity which is theoretic! attainable in a steam jet is, therefore, enormous, and this causes extraordinary mechanical difficulties. For if we want to utilise the whole of this energy., by allowing the jet of steam to impinge on the blades or buckets of a turbine wheel, simple math: ! considerations show that the velocity of the blades should not be short of' one-half the velocity ol the or maximum efficiency we should have a speed of the order of 100 metres/sec. at the periphery of the wheel which carries the blades. The number of revolutions per minute ; sary to produce such peripheral speeds for a wheel of moderate dimensions reaches the value of about 30,000 and such speeds are impracticable for two reasons. First, th no construc- tional materials which can withstand such enormous peripheral speeds. The rotating parts, would fly to pieces on account of the large centrifugal forces developed and there would be mechanical vibration of the rotating shaft. Secondly, if the shaft is to be coupled to another machine which should usually rotate at a much, lower speed, say 2000 r.p.m., the problems of gearing become almost insuper- able. These difficulties were first successfully overcome by De Laval for low-power turbines. For the gearing down, he used doublc-hel ! • wheels with teeth of specially fine pitch. To enable the requisite peripheral speeds to be obtained' with safety, the turbine wheel was made very thick in the neighbourhood of the axis (see Fig. 19) so that it can withstand the stresses set up by rotation and further tile shaft made thin. A De Laval turbine is illustrated in Fig. 20. The steam jet issuing from the fixed nozzles impinges on the blades, has its direction Fig. 19.— Vertical section wheel of De Laval turbine. 232 HEAT ENGINES CHAP. changed by them and thereafter escapes to the exhaust. Due to this action of the steam the blades are set in rota- tion. These are known as 'impulse turbines' be- cause the steam in pass- ing through them is not accelerated but has only its direction changed. The interaction of steam with the blades is clear ly indicated in Fig. 21. In these turbines the bli are 'parallel*, i.e., a passage between two blades has nearly the same cross-sect everywhere so that the blades offer very little obstruction to the steam jet and hence the p sure of the steam re- mains practically the same on both sides • of the wheel. Fig. 20.- -A De Laval ti 30. Division of tfie Pressure Drop into Stages. Rateau and Zoily Turbines. — As a prime mover, the De Laval turbine is quite efficient for producing genera- ed steam must be used so. that the j sure drop in the would be enormous, and ice for maximum effi- ciency the blades must move with very high velocity. .Such large velocities are incompati- ble with safety. For this reason, the entire pressure drop is divided into a number of stages. Each stage consists of one set of nozzles and one ring of moving blades mounted on a wheel fixed to the shaft. Across each - the pressure drop is small so that the blade velocity necessary for maximum efficiency is considerably reduced and can be easily attained in practice. This principle of pressure compounding was first given by Parsons and later adopted by Curtis, Rateau and Zolly to the De Laval type. Rateau turbine has 20 to 30 stages. Zolly has about 10. l r -!. 21, — Action of steatn on the blade* a De Laval turbine. IX.] REACTION TUUB1N£S m Steam 31. Reaction Turbines— Parsons' Work.— Parsons* began his re- searches on steam turbine about the same time as De Laval (1884) » but he did not accept the fetter's solution of the problem, and went on with his own investigations which were completed by about 1897. His researches resulted in the evolution of a completely ; i — the so-called Reaction Turbines. The mechanism of this type is shown in Fig. 22* The essential feature of the Parsons' turbine is its blading system. This consists of alternate rings of a set of fixed blades mounted on the inside o£ the cylindrical ease or the stator, with a set of moving blades mounted on the shaft or the rotor, arranged between each pair of rings of fixed blades as shown in Fig-. 5 There are no separate nozzles for producing a jet, but the fixed blades ace as nozzles. It will be noticed that the blades have convergent passages, i.e., the cross-section of the passage goes on decreasing towards die exit side. the s t e a m passes through the moving' blades, is a drop of pressure in equence of which the steam jet increases in velocity as it traverses the blade- age, and as it issues out, a backward reaction is produced on the blades which conse- Hill Fdhk Moving Btadea "Fixed Blades Moving Bkdes JJJJJ> JJJJJJ- Fifi. 22.- -Tbe blading- _ system M turbine. of quently begin to revolve. The steam then "again passes to the next set. of fixed blades where it again acquires velocity and enters the next set of moving blades nearly perpendicularly. The moving blades are secured in grooves on the rotor shaft with their lengths radially outwards and parallel to each other. The fixed blades are secured' in grooves in the enclosing- cylinder and project inwards almost touching the surface of the spindle. The steam is admitted parallel to the axis of the shaft (axial flow type) . One pair of rings of moving and fixed blades is said to form a stage, and .generally a turbine has a large number oE stages. Pressure Compounding. — On account of die peculiar form of the blade-passages, the pressure goes on falling rather slowly, so that hole pressure drop is spread over a large number of stages which may amount to 45 or even 100. In consequence of this arrangement the velocity of the jet is very small compared to that of the De Laval type and the requisite blade speed is 100 to 300 ft. per sec. * Charles Algernon Parsons (1854-1931) built the fir<L reaction turbine in 1886 at Newcaslle-on-Tyne. He was awarded the Copley & the Rumford medals ot the Royal Society for hh invwitions. 234 HEAT ENGINES CHAP, This reduction of velocity brought about by the distribution of avail- able pressure over a large number of stages, constituted the master- invention of Parsons which was also applied to other turbines with great advantage as already mentioned. We have stated above that the steam enters the moving- blades perpendicularly. _ In actual practice, however, the steam enters the moving blades with some relative velocity giving it an impulse as in De Laval's turbine and so the turbine works partly by impulse and partly by reaction. Thus it will he seen that the distinction between 'impulse' and 'reaction' turbines is more popular than scientific and should not be taken too literally. All turbines are driven by 'reac- tion/ due to the alteration of velocity of the steam jet in magnitude and direction. The real distinction between the two types is that in the 'impulse' turbines the reaction is due to the steam jet being slowed down while in the pure 'reaction' turbine the reaction is pro- duced by the acceleration imparted to the steam in the g blades themselves on account of the pressure drop. Still, however, by simply looking at the blades one can say whether the turbii the 'impulse* or 'reaction' type, the impulse blading being charaete rised by parallel passages and the reaction blading by the convergent passages. During the last thirty years high-pressure ga . i: been uti Used in place of steam to drive the blades of a turbine. In the inter. nal combustion gas turbine the combustion of some oil serves to heat sed air to a high temperature and high pressure, arid this high pressure gas drives the turbine wheel. 32. Alternative types of Engines. — During the Second World War Chi developed in Germany and Italy mainly with the idea of devising a powerful means of propulsion for >speed aircraft. In a modern turbo-jet a mixture of air and fuel ticked in and compressed by a compressor and then ignited so as are. rhis high pressure gas traverses a ti bine where it gets accelerated and the pressure falls, the gas escap- ing at the rear o[ the machine with a high jet velocity, The machine on account of the reaction thus furnishedj move yard with tre- mendous velocity. In this way the modern jet plane has been able to attain supersonic velocities le< velocities greater than the velocity of sound. Mention may be made of the Rocket and the guided missile whose use by the Germans in World War II as V-2 rocket caught world imagination and whose potentialities for space travel are bou to make it an important tool of the future for scientific research. The essential difference between a rocket and a jet aft is that the Cornier carries its own oxygen supply and thus can operate in vacuum while the latter obtains oxygen from the air. The propellant. solid or liquid, ignites producing a jet and the rocket takes off. The final rocket velocity V r — zv log f M* where n> is tlje jet velocity and M* denotes the ratio of the initial mass of the rocket to the final mass after combustion. In the two-stage rocket a smaller second IX. TI HEMODYNAMICS OP REFRIGERATION 235 rocket is placed hi the nose which ignites a lew minutes after take- off and the second rocket detaches itself and forges ahead under its own power with a tremendous velocity. Such a multi-stag was used by the Russians to launch the first artificial earth sal on 4th October, 1957 and the first artificial planet on 2nd January, 1959- The thrust developed by these rockets is enormous. THERMODYNAMICS OF REFRIGERATION 33, Tn Chapter VI we described several refrigerating machines and considered the general principles upon which their action depends. We shall now investigate the problem of die efficiency of these machines. In order to compare the efficiency o£ the various machines we shall introduce a quantity called the coefficient of pe: nee of a refrigerating machine. If by the expenditure of work W a is deprived of its heat by an amount q, its coefficient of performance is given by q/W, Tn Section 15 we saw that if a Carnot engine with a perfect gas as the working substance is worked backwards between the source of heat at T] and the condenser at T.,, the working substance abs- tracts an amount of heat (X from the condenser and yields (h to the source where Q^ and Q* are connected by the relation Qt/Ti = Qg/Tjf The work done in driving the engine is Ql — Qz- Hence the coefficient of performance of a Carnot engine working refrigerating machine is a. r. di - Qa " ** - T f ' (20) It can now be easily proved much in the same way as on p. . that no refrigerating machine can have a coefficient of performance greater than that of a perfectly reversible machine working the same two temperatures, and that all reversible refrigerating machines working between the same temperatures have the same coefficient of performance. Tt thus follows that the coefficient of performance of any perfectly reversible refrigerating machine working between temperatures T 1 and T- 2 , whatever the nature of the working substance, is equal to T 2 / (7*i - T 2 ), and this is the maximum possible. A Carnot engine using perfect gas and working backwards will be a most efficient refrigerating machine but its output will be due to the small specil heat of gases. For large capacity we must employ a liquid of large latent heat of vaporization and allow it to evaporate at the lower temperature. We have seen in §6, p. 128 how practical considerations have led us to select ammonia and a few other substances for this purpose. 34. Efficiency of a Vapour Compression Machine.-— We shall now calculate the efficiency of the vapour compression machine HEAT ENGINES [OJAP, l\ ; E A ^T Fig. 23. — Cycle of a vapour compression machine. described on p> 127. Let us trace the cycle of changes which the working substance undergoes. For this purpose assume the valve V to be replaced by an expansion cylinder. AB (Fig, 23) represents the evaporation of the liquid in the refrigerator, BG the adiabatic compression of the vapour by the compressor, CD the liquefaction of the sufjsLance inside the con- denser and DA the adiabatic ex- pansion of the liquid in the expan- sion chamber (not shown) . The cycle ABCD is perfectly reversible and hence its coefficient of perfor- mance is given by equation (20) . In actual machines the throttle valve takes the place of the expansion q f Iinder* This is done for mechanical simplicity. Conse- quently the part DA of the ideal cycle is replaced by DEA, On account of die irreversible expansion through the valve there is a loss of efficiency. The losses are twofold. An amount of work equal to the area DEA is lost and there is also reduced refrigeration effect since some liquid evaporates before reaching the refrigerator. The coefficient of pei .: for the actual machine can be calculated as the case of Rank' le (n, 220) from the values of the total of the working sub; the different states. Books Recommended. i . \e, Engines. fug, Steam Engine and other Heat En Glazebrook. A Dictionary of Applied Physics, VoT. 1., Articles on Steam Engine ; Engines, Thermodynamics of Internal Combus- tion : Turbines., Development of ; Turbines, Physics of. 4, Judge., Handbook of Aircraft Engines (1945) , Chapman & Hall Ltd. .1, Clarke, Interplanetary Flight (1952), Temple Press Ltd., London. CHAPTER X THERMODYNAMICS 1. Scope of Thermodynamics. — ■ The object of the science of JTeinL is the stud) p£ all natural phenomena in which heat plays the leading part. We have hitherto studied a few of these phenomena in a detached way, viz./ properties of gases, change of state, varia- tion of heat, content, of a body (calorimetry) , There are other pheno- mena (e.g*., Radiarion, i.e., the production of light by heated bodies) which have not yet been treated ; there are others which, strictly speaking, do not come under physics, vis,. Chemical equilibria, Elec- tro-chemistry, but in which heat plays a very prominent part. The detailed study of all these phenomena is beyond the scope of this volume. The object of this chapter is simply to develo] methods for the study of all such phenomena. The first requisite for such studies is the definition of the thermal state of a body or system of bodies. Next comes the development of general principles* The studies fall under two heads-*- (a) the study of energy relationships, (it) the study of the direction in which changes take pi ace t The guiding" principles for the two cases are the First and the Second Laws of Thermodynamics. 2. The Thermal State of a Body or System of Bodies.— The I mal state of a simple hoi ius body like a gas or a solid is defined by its temperature 7\ pressure p or the volume V, These quantities are, therefore, called Tkermodynamical Variables. Of the three quantities only two are independent, the third being automa- tically fixed by the value of the first two, for it is a common obser- vation that the pressure, volume or temperature of a substance in any state, solid, liquid or gas, is perfectly definite when the other two quantities are known. Hence for every substance the pressure, volume and temperature are connected by a relation of the form P = ffaT) (1) This is called the 'Equation of State' of the substance. The oE (1) is known only for a perfect gas (Chap. IV), But though (I) is erally unknown we can proceed a good deal towards study- ins the behaviour of a substance under different conditions by using differential expressions. In the subsequent work we shall occasionally use differential calculus and some of its important, theorems. The reader must clearly understand the physical interpretation of these and for that purpose the following- brief mathematical notes are appended, 3. Mathematical Notes.— Suppose that y is some function of the variable x and is continuous over a certain range of x. Let 8x be some increment in the value of x and let Sy be the corresponding 238 THERMODYNAMICS [CHAP. Increment of y, such that y -j- &y still lies within the range of y con- tinuous* Then the value of the ratio £ as fix tends to zero is the ox differencial coefficient of y with respect to x and is written -r- . ax If x is a function of the two variables y and z., ue. t * = / <y, * there are two differential coefficients of the function. In the first case the function may be differentiated with respect to 5? keeping z constant, and in the second it may be differentiated with respect to ping y constant. These differential coefficients are known as parlial differential coefficients and are written ».- I where the suffix outside the bracket denotes the variable which is kept constant during differentiation. In order to find the variation in the quantity x when y and z are both increased by the amounts Sy and &z we must make a double application of Taylor's theorem. Now if x=f(y) we have from or's theorem (3*1 _ When x is a function of the two variables y and z, the value of ., z + &) may be obtained by first considering the second varj it z-f 8z and expanding in terms of Sy only teorem. In the second stage y is kept constant and the _ Ll - l) h are functions of z-f- Sz are expanded by Taylo >rem in Ll - rii Let us denote the partial derivative of f(j, z) with respect to v by /., and so on. Hence =f[J\ .--3:} H >, e+fc) KW- .,G?{.y 5 c-f-S^r) + higher terms in tj>. Expanding each term on the right-hand side by Taylor's theorem in terms of S2 we obtain +H<W/wCa *) +28>8c/, 4 o-, «) + (&e)v«t>!» z)} + . (3) If instead of expanding first in terms of Sv, we had expanded first m terms of 82 and then in terms of 8y we would have obtained an expression similar to (3) with the only difference that instead t y t iy t z) Id now obtain jL , (y t z)> Since these two values of Ar-f-Sx must be identical we have MATHEMATICAL NOTES or (*) i _ d~'.-: ie order in which the differentiation is performed is immab If the variations 8y, gz are continuously decreased so that the terms involving their higher powers may be neglected, equation (3) reduces to the form -By-/, +&£./( or &* S* is the theorem of partial differentiation. Let us consider the equation (5) (6) On comparison with (5) we see that if equation (6) is obtained from an equation of the form (2) we should have M !."-(&■ (7) Conversely, iF M and A r are of this form there must exist a functional relalion of the form (2) connecting the variables y and & When (2) exists condition (4) h satisfied which from (7) is equivalent to the condition (8) Bz Hence if condition (8) exists, equation (6) must result from an equation of the form (2) . Then Sx is said to be a perfect differential. The physical meaning of this is that the change in x for the change in the variables y and z from y L , z x to y 2 , z* is given by *i - x 2 = f(y t , Zt) - /(V 2 . ZsO , and is consequently independent of the path of transformation, depending only upon the initial and final values of the variables. There exist some physical processes for which a relation of the type (6) exists but (8) is not satisfied. Then §x is not a perfect liferent ial. As a mathematical example consider the expression 2 Then Sx « 3k dy ■ V 2fd& dx „ Bx a dj'dz dzdy m (10) -= 240 Tl I FRMODYNAMI CS [CHAP. e (9) cannot be integrated as can be easily verified, The condi- tion of integrability is just the reverse of (10) . If we multiply" (9) by yz the resulting function 3y 2 z*dy -j- 2y ? 'zdz = dfy^z 2 ) is integrable, The quantity yz is called the In --fating factor of the expression In equation (5), suppose z is kept constant, i,e. f §2 = 0. '1" on division by $x, we obtain or 1 UAUd, which implies that the reciprocal of tile partial differential coefficient is equal to the inverted coefficient 4. Some Physical Applications. — From equation (1) we have, on differentiation, in the general case *-(&),■*+{&).": (12) For an i. so baric or isopiestic process, dp = 0. Hence equation (12) yields dp_ dT ) e U'/tU' > /a H ,-r-) —coefficient of volume expansion in an iso- o 1 I p baric process (Chap. VII). v 1^7) = bulk modulus of elasticity in an isothermal process, or inverse of the coefficient of if nx:?. nihility. — l-^-l = coefficient, of pressure increase in. ar choric (constant volume) process. ^Equation (13) shows that the volume elasticity, the volume co- efficient of expansion and the pressure coefficient of expansion are not independent but are inter-related. As an illustration, consider mercury at 0°C. and under atmos- pheric pressure. The coefficient of volume expansion is LSI >< JO -4 per °C. and the coefficient of compressibility" per atmosphere is 3.9 X 10-* Hence from (13) { m 1 Tdv ] v IdTl \of ! v l dp \ T cr = 4b -5 atm.. 3»9xl0-< x.] DIFFERENT FORMS OF ENliRGY 241 and the pressure coefficient 7fe*7." 46-5. Hence it follows that a pressure of about 46 atm. is required to keep the volume of mercury constant when its temperature is raised from Q°C« to PC. Thus if the capillary tube of a mercury thermometer is just filled with mercury at 30 °G., the pressure exerted in the capillary when it is heated to 34 °G. will become 4 X 46\5 ^= 186 atmospheres. 5. Different Forms of Energy, — As thermodynamics is largely a study of energy-relationships, we must clearly understand the term 'Energy'. The student must be familiar with the concept of 'Energy' from his study of mechanics. 'Energy' of a body may be denned as its capacity for doing work and is measured in ergs on the C.G.S. system of units. Everybody must be familiar with the two ordinary forms of energy, viz., the kinetic energy and the potential energy. A revolving fly- wheel possesses a large amount of kinetic energy which can be utilised in raising a piece of stone from the ground, Le,, in doing work. Tn this process the kinetic energy of the flywheel is converted into the potential energy of the mass of stone. The mass of stone also in its raised level possesses energy which we call potential energy and which can also be made to yield work. For when the stone falls down under gravity the flywheel can be coupled to a machine and be made Lo do work. Thus kinetic and potential ener- gies are mutually convertible and in a conservative system of forces the sum of the kinetic and potential energies is constant throughout the process. All moving bodies possess kinetic energy by virtue ol their motion and all bodies placed at a distance from a centre of force possess potential energy by virtue of their position. There are other forms of energy also with which the student ol physio must have become familiar, and which we shall now take up. We have already pointed out that heat consists in the motion of molecules and must therefore be a form of energy. The experi- ments of Joule and Rowland showed that there exists an exact pro- portionality between mechanical energy spent and the heat developed. Friction is the chief method of converting mechanical energy into heat. The conversion of heat into work has also been accomplished by the help of steam engines and other heat engines which have been completely described in Chap. TX. Thus it has been fully established that heat is a form of energy. ^ Energy ma y be manifested in electrical phenomena also in a variety of ways. It is well known that when two current-carrying coils are placed with their planes parallel to each other, they attract or repel each other depending upon the direction of flow of the current. In this case we may take the more familiar example of the tramway. Here electric currents are made to run the carriage. Again 16 242 THERMODYNAMICS [CHAP, -a wire carrying a current becomes heated Tints electricity is also a form of energy which can be easily converted into mechanical or thermal form." Energy may also be manifested as chemical energy. For we know that a bullet fired from a gun possesses enormous kinetic en< which is derived from the chemical actions that take place when the powder detonates. Further a large amount of heat is evolved during a number of chemical reactions, viz., in the combination of hydrogen and oxygen. Also the chemical aciions going on in a voltaic cell produce electrical current. Thus chemical energy can be easily con- ed into mechanical energy, heat, electrical energy, etc. There are various other forms of energy, all of which can be converted into some one of the other forms. The important ones are the following ;— magnetic energy, radiant energy (Chap XI) , surface energy (surface tension), radioactive energy and energy of gravitation. There are three important laws governing- energy transforma- tion :— (I) The Transmutation of Energy, (2) The Conservation of Energy, 0) The Degradation of Energy, 0. The Transmutation of Energy. — This principle states that energv converted from one form to another, We have already- seen how mechanical energy, electrical energy and chemical energy can be converted into heat. In fact all kinds of energy can be directly .verted into heat. Mechanical energv obtained from a water-fall irking a turbine, may be utilised to generate electrical energy by Uphng the turbine to a dynamo. Electrical energv may be con- rted into thermal as in electrical furnaces, or into' chemical ?rgy as in t: secondary cells. Heat energy can be ced into electrical energy as in the phenomena of thermo-electri- n to mechanical energy ; ces r and so on. Examples couM be multiplied indefinitely. The Conservation of Energy,— The principle states that during • interchanges of energy, the total energv of the tem remains constant. In other words energy is indestructible and unbeatable by any process. No formal proof of the law can be given. It is a generalisation based upon human experience and is amply home out by its consequences. The law arose out of the attempts of earlier philosophers to devise a machine which will do work without expenditure of energy. They however railed in their attempt- Later it was recognized that al! the machines invented were simplv devices to multiply force and could not multiply energy, and the true func- tion of n machine was to transmit work, Onlv that much work could he obtained from a machine as was supplied to it, and that too in idea cases. This shows that energy obeys the law of conservation. In these cases, however, all energy cannot be recovered as some is dissipated as heat But heat is also a form of energy, and on equating the work obtained plus the heat developed to the\vork done, we find that the relation of equality is satisfied. Thus the principle is estab- led for this case and follows from a result of human experience which may be stated m this form: x.] FIRST LAW OF THERMODYNAMICS 243 "It is impossible to design a m-it-chine which will create energy out of nothing and produce perpetual motion. Energy can only be transformed from one form to another" 8. The Dissipation of Energy. — Energy, we have seen, is capable of existing in different forms. In some forms however it is more available to us than in other forms, Le t> we can get more work from energy in one form than in another. For example, mechanical energy is the highest available form while the heat energy of a body at a low temperature is in a much less available form. ' Now the law of degradation or dissipation of energy says that energy always tends to pass from a more available to a less available form. This is called the degradation or dissipation of energy, and is intimately connected with the Second Law of Thermodynamics ; it will therefore be taken up later. FIRST LAW OF THERMODYNAMICS 9» The First Law* — We have alreadv enunciated the law of con- servation of energy. The First Law of Thermodynamics is simply a particular case of this general principle when one of the forms of energy is heat It may be mathematically stated in the form* sc> = w -f- m, (14) where SQ is the heat absorbed by the system, S£/ the increase in its internal energy and SW the amount of external work done by it. It may be remarked that 3Q is here expressed in energy units, and can be found by multiplying the heat added in calories by L the mechanical equivalent of heat. For a simple gas expanding against an external pressure p, SW = p8V, but for surfaces, magnetic bodies, etc, other suitable terms are to be added to take into account other forms of energy such as mechanical, electrical etc. The quantity 'internal energy' is defined with the help of the principle of conservation of energy by considering a thermally insulated system for which 8U = -&W. Changes in internal energy can thus be measured in terms of the external work done on the system. Applications of the First Law 10. Specific Heat of a Body. — It is easy to see that the internal energy of a body or a therm odynamical system depends entirely on its thermal state and is Uniquely given, by the independent thermo* dynamical coordinates. In the case of a simple homogeneous body we have seen on p. 237 that, any two of the variables p, v f T are sufficient to define its state uniquely. Choosing therefore V and T as independent variables we have U = f{V t T)> . * It was first stated in this form by Gausius. • OS) 244 THERMODYNAMICS [CHAP. where U and V refer to a gram-molecule of the substance. Differen- tiating (15) we get* f -(#).«■+ (w) r * ■ • • ™ For a perfect gas LI —0 (p. 47); for a gas obeying van der WaaJs* equation (see § 33) IBU\ _ a ( 3 V I T F- • l£ an amount of heat 8.Q be added to a thermodynamical system* say a perfect gas, part of it goes to increase the internal energy of the gas by dV, while the remainder is spent in doing external work, e.g., when the gas expands by a volume dV against the pressure p. We have from the first law, 8Q=idUi-pdV t , (17) where SQ, dU are expressed in energy units. Substituting for dU from (16) in (17), we get and dividing by dT we have Hl_ldU\ r idu\ ]dV SO Now ■ j^= C, the gram-molecular specific heat. If volume be kept constant, (#).-(#).-*■ • - - w from definition. If pressure be kept constant, I ~) =C b , the V dT f p specific heat at constant pressure. Hence For a perfect gas { ?£) ^ -0 and p (|^l) = E, hence using (18) *It is thus seen that the quantity dV is a [perfect differential of a function U at the variables dcfinmjf the slate of the bodv; we can therefore write dV place of SU. _ Tn contrast with this the quantities, SQ, 8lV are not perfect ■ seen from the indicator diagrams since differentials. 2 ."j 1 , 3 f" d^P*™ 1 "Pan the path of the transformation and not only upon the initial and Jmal states. Hence we have denoted these changes by $Q and &W Some people use 4 in place of these 3's hut then this d should "not" he taken It J e d ?ff crcntial scnsc - *" is however a perfect differential and we can write v\¥ = paV. WORK DONE IN CKKTAIN PROCESSES 245 C p -C; = R, ... * , (20) as has been deduced on p. '16. The relation for solids and liquids is complicated (see sec. 38, Example 1) . It is easy to see that 8Q, the heal added, cannot be deter- mined only from the inilial and final states of the body, and a ledge of how the heat, has been added is essential. Hence C, the specific heat, has no significance unless the external conditions are prescribed. We have defined C p and C , but there may be other processes, e.g., adiabatic, when §(> = and C = 0. 11. Work done in Certain Processes. — The following expressions for work done in different processes have been already proved : — (a) Work done by a perfect gas in isothermal expansion is (vide p. 215) equal to RTlQg t (V & IV'.). (b) Work done by a perfect gas in adiabatic expansion in which the temperature falls from 7\ to T 2 (vide p. 215) y— 1 (C) 12. In the case of a cyclic process the substance returns to its initial state and hence dU = 0. Equation (17) then yields dQ_ = pdV/j, i.e., the heat absorbed by the substance is equal to die external work done by it' during the cyclic process. Thus in Carnot's cycle the work done during a cycle is Ql - (£., Discontinuous Changes in Energy — Latent Heat.— When a body is in the condensed state (solid or liquid) and is subjected to increasing temperature, the state may suddenly change from solid to liquid form (fusion), or from liquid to gaseous form (evaporation) , or from one crystalline form to another (allotropic modification) . In such discontinuous changes the energy also changes discontinuously (phenomena of latent heat). Let us consider the evaporation of 1 gram of water at 100°G, and 1 atuL pressure. The application of First Law yields L = u 2 -u 1 -\-p{v z -v 1 ), (21) where %, v 2 respectively denote the internal .energy and volume of 1 gram of vapour, and %, Vj_ the corresponding quantities for the liquid. Equation (21) expresses the fact that the heat added is spent in two stages : (1) in converting 1 gram of water to 1 gram of vapour having tile same volume, (2) in doing external work, whereby the vapour produced expands under constant pressure to its specific volume. 246 ISKRMODYNAMICS ; CHAP. u^ — Ui is sometimes known as the internal latent heat and p(v s — Vj) as the external latent heat. For water at 100°C, Vi = 1 tc, v 2 = 1674 c.c. (specific volume of saturated vapour) , „ .. . 1-013 xlO»x 1673 . ..„ , Hence p(v t - v{\ — n-= — ^xi cal. sz 40.5 cal. 1 N ' 4-lBx 10 7 Since L = 538.7, the Internal Latent Heat « 2 - «i = 498.2 Cal, It is often convenient to denote the different states of aggregation of the same substance according to the following convention, [H 3 0], (H 2 0), H a O denote water in the solid, liquid and gaseous states respectively. The quantity taken is always a mol. unless otherwise stated. The symbolical equation H a O = (H a O) -J- 10,170 cal ... (22) expresses the experimentally observed fact that the energy content, of a mol. of H 2 (gas) at 0°G exceeds the energy-content of a mol o£ H.O (liquid) at'0°C. by 10,170 calories, die change taking place at constant volume. We have further (H 2 0) = [HaO] + 1^30 cal, . . (23) us, t energy of a mol. of liquid II s O at 0°C. exceeds the energy of a moL of ice at 0°C. by 1,430 cal From equations (22) and (23) we have by addition H s O = [H..O] + 11,600 cal. i.e., heat of sublimation of 1 mol. of ice at constant volume is 11,600 calories. This is a consequence of the First Law applied to a physical process and can be easily verified, 13. The First Law applied to Chemical Reaction* — Hess's Law of Constant Heat-sommatioiL-— The First Law can be extended to Chemi- cal Reactions, Thus when 2 mols, of H 2 and 1 moL of 2 are exploded together in a bomb calorimeter, 2 mols. of (H a O) are formed and 136,800 calories of heat are t-volved. Allowing [or the heat evolved in passing from the gaseous to the liquid state, we obtain from the First Law, since SQ = - 116,500 and BW =0, 2t/ HQ + Po.-aOW.o = H6,500 cab !>.., the energy of 2 mols. of H-. and 1 mol. of Or, exceeds the energy of 2 mols, of 'H a O by 116,500 calories, Hess stated in 1840, before the First Law had been discov i : that if a reaction proceeds directly from state 1 Co state 2, and again through a seri: o£ intermediate states then the heat evolved in the direct change is equal to the algebraic sum of the heats of reaction in the intermediate stages. This is an important law in Thermo, chemistry and is known as Hess's Law of Constant Heat Summation. The law is illustrated by the following examples : — G (diamond) + O a = C0 2 -f 94,400 cal CO+ £O s = CO a + 68,300 caL x. SECOND LAW OF THERMODYNAMICS 241 We have by adding C (diamond) + 2 = CO s -J- 94,400 cal. This is verified by experiments. We take another example. Let the symbol (LiOH) m denote the solution : ol. of the substance. Then take the equations:— 2 [Li] 4- 2 (H 2 0) = 2 (LiOH) an -f H s + 106,400 2[LiH] -f 2 (H s O) = 2 (LiOH)*, -f 2H 2 + 63,200 The energy evolved is very easily determined from solution ex- periments. Subtracting the lower equation from the upper one • obtain 2[Lil -f H* = 2[LiH] +43/200 cal. i.e., the heat of formation of two mols of [LiH] from [Li] and T-l 2 is 43,200 cal, THE SECOND LAW OF THERMODYNAMICS AND ENTROPY 14. Scope of the Second Law.— The Second Law of Thermodyna- mics deals with a question which is not at all covered by the First Law, viz., the question of the direction in which an}' physical chemical process involving energy changes takes place. A few illustrations will clear the point. Let us consider tru thermodynamical process illustrated by the symbolic equation :— 2H 2 -f- 2 = 2H 2 + U calories. The equation tells us that if 2 mols. of H 2 gas combine with o mol of O s , 2 mols of H.O vapour are formed and U calories of heat are evolved, or vice versa, when 2 mols of H 2 Q-vapour are decompa completely into H ; , and 2 , U calories of heat must, be a', irh This result is obtained as a matter of experience from ealorimetric experiments, and we interpret the result according to the First Law of Thermodynamics by saying that in the combination of 2 molecu of hydrogen with 1 molecule" of oxygen to form 2 molecules of H s O, [Tie diminution, in energy amounts to JU/N ergs. This can be further utilised in calculating energy relations in other reactions. But the first law cannot tell us in which direction the reaction will take place. If we have a mixture consisting of H 2J O a and H a O-vaponr in arbitrary proportions at some definite temperature and pressure, i lie first law cannot tell us whether some H 2 and Q 2 will combine to form H ; .0 or some H 2 will dissociate into H 2 and Q 2 , the system i hereby passing to a state of greater stability. Or we can take a physical example. When two bodies, A and B. exchange heat, the first law tells us that the heat Iosl by one body is equal to the heat gained by the other. But it does not tell us in which direction this heat will flow. Only experience tells us that heat will pass from the hotter body to the colder one spontaneously. 248 THlUtMODVNAMlCS [CHAP. but in speaking of hotter and colder bodies we are making use of a V physical concept (temperature) which is not at all included with- in die scope of the first law. Tor giving us guidance in deciding the question of direction in which a process will take place, we require a new principle and this principle, which arose out of Carnot's specu- lations about the convertibility of heat to work, is genially known as the Second Law of Thermodynamics. 15. Preliminary Statement of the Second Law. — The problem of convertibility of. heat to work has been treated in Chap, IX. It is readily observed that the main question there was about the direction of energy transformation. Mechanical energy and heat are only different forms of energy, but while mechanical energy can be com- pletely converted to heat by such processes as friction, it is not possible to convert heat completely to work. Even by using reversible engines which are the most efficient, only a fraction can be converted to work. The conversion is therefore only partial. The question is, why is it so ? An answer to this question has been given already in Chap. IX, viz., if it were possible to design an engine more efficient than a reversible engine, we could continuously "convert heat to work, and this will produce perpetual motion of a kind. We may call (his perpetual motion of the second kind. We are convinced that this is not possible, and we may start from a statement which expresses our con vie Lion just as the first law is based on the conviction that Energy cannot be created, but can only be transformed. We may ateirfent In the following form: — . impossible to construct a Heat Engine which will con- ly abstract heat from a single body, mid 'convert the whole of :he working system," This ement is equivalent to Clausius or Lord Kelvin's" statement of die Second Law. Lord Kelvin stated the law in this, form : — '7* & impossible by / inanimate material agency to derive mechanical effect from ■ lion of matter by cooling it below the temperature of the lest of the surrounding objects," Clausius staled it in the form : — "It is impossible for a self-acting machine, unaided by any external agency, to convey heat from one body to another at a higher temperature,' or heat cannot of itself pass from a colder to a warmer body." It must be clearly understood that these statements refer To con- tinuous or cyclical processes. For it is always possible to ]et heat pass from a colder to a hotter body and at the same time the sub- stance may change to a state different from its in i tin! one. As an example consider an isolated system comprised of a cold compressed gas and a hot rarefied gas communicating with each other through a movable piston. When the piston is released, the hot vapour is com- pressed and heat is thereby transferred from the cold to the hot gas. The state of the two gases however changes as a result of the process. To face p. Z4S Lord Kelvin (p. 248) William Thomson, Lord Kelvin of Largs, born on June 26, 1824, in Belfast died qn December 17, 1907, His important contributions to Heat arc the Second Law, the absolute thermometric scale and the Joule-Kelvin effect. He carried on valuable researches in Electricitv. Rudolf Cuaustus (1822 (p. 248) at KosKn, he studied in Berlin and became Professor of Physics successively at Zurich, Wureburg and Bonn. Simultaneously with Lord Kelvin he announced tJie Second Law of Thermo- dynamics. He was one of the founders of the Kinetic Theory of Gases. X.] ABSOLUTE SCALE OF TEMPERATURE 219 16. Absolute Scale of Temperature.— Lord Kelvin showed in 1881 that with the aid of the ideal Camot engine it was possible to define temperature in terms of energy, and the scale so obtained is inde- pendent of the nature of any particular substance, £ Kelvin started the ret.uk (p. 219) that' the efficiency of all reversible engines ing between two temperatures is a function of the two tem- peratures only and is independent of the nature of the working sub- stance. This may be mathematically stated in the form W Q* -/(*!> ^ (24) where W is the work done by the Camot engine, Qt the heat taken in and i3 6 2 tile temperatures* between which the engine works. Now W= Qi-C&f where Q* is the heat rejected. Hence ^ =/&, *■). or ft. - F(e lt e t ), where F denotes some other function of L and Q%.„ It can easily shown that the ratio Qi/Qa can be expressed in the form «/f(0i)/^ (&i) ■ -"For if we have reversible engines working between the pairs of temperatures (0 a , d b ), {$$, 3 ), (B u 3 ) and the heat absorbed or evolved (as the ease may be) at these temperatures be <b> Gap gs, then ^— m, 0*}, Q n. ^ = FiB,, 6,), £- - F(0 1; 3 ). . (25) ft. ^ *" ft* < From these by multiplying the first two expressions we get (L = F(B lt fl B ) X F(B tl 3 ), Hence the function F must satisfy the relation F\9 X , 9 a ) = F(9 U 9 t ) X F%, This will be so only when F(Q it d- £ ) is of the form ^(0, )/${#, fore for any reversible engine ft* *&) " ' It will be seen that i| & 1 >tf 2( C>]>f.> 2 and therefore 'P(0 1 )>'ii(6 i ) ) i.e., i/>(0) is a quantity which increases monotonically with Q and may be used to measure temperature. Denote the value or ip (0) by t. The relation (27) then becomes 0,1 *i (26) There- (27) & (28) * These may be measured on any arbitrary scale. 250 TJIERMODYNAM ICS CHAP, Equation (28) is used to define a new scale o£ temperature t which is tailed the absolute or the thermodynamic scale. This scale does not depend upon the properties of any particular substance for equa- tion (28) is universally true. The ratio of any two temperatures on this scale is equal to the ratio of the heat, taken in and neat rejected by an engine working reversibly between the two temperature. 1 ;. The zero of this scale (r=0) is that temperature at which (from 28) and hence Ti r — (^, Thus all the heat taken by the engine has been converted into'work and the efficiency of the engine is unity, t cannot be less than this, i.e., negative, for if it were so, Q 2 would be negative which implies that the engine would be drawing heat both from the source and the sink. This is impos- sible from the second law and hence t = is the lowest temperature conceivable. This is the absolute zero of temperature which need not be the temperature at which all molecular motions cease to exist as one might otherwise infer from elementary kinetic theory. The zero having been determined, let us fix the size of the degree.. In conformity with general usage we suppose that the interval be- tween the freezing and the boiling points of water is divided into 100 equal parts on this scale, viz, f 3*. h-ioo (29) The thermodynamic scale has thus been completely defined and fixed. scale can be actually realised in practice. For this we must investigate the property of some actual sub- stance. A perfect gas is very suitable for this' purpose. In fact we have already shown on p. 216 that for a Carnot engine using a perfect gas as the working substance, . (SO) where T lf T z are the temperatures measured on the perfect gas scale. Equation (28) combined with (30) yields (81) -*", '.:.. the ratios of any two temperatures on the thermodynamic and the perfect gas scale aTe equal. Thus the absolute narnic scale coincides with Che zero oC the per!) , ■ -■. ther- mometer; for if t o — O, r 2 = 0. Further from (29) and (31) since the Interval between the ice point and steam point is 100° on both scales, r, e , and T iM are identical, and in general from (31) other temperatures on the two scales are also identical, Thus the thermo- dynamic scale is given by the perfect gas thermometer,. For the methods of reducing the ordinary gas scale to perfect gas scale p. & x.) ENTROPY ENTROPY 251 17, Definition of Entropy.— The formulation of the Second Law, as on p. 248, is not of much use for finding out the direction of change in a chemical or physical process. We want a more general and mathematical formulation which was supplied by Clausius. Clausius shifted the interest from the problem of convertibility of heat to work, to that of change of thermal state of the working substance, for it is only the working substance which undergoes a thermodynamical change in the process. He observed that when the heat Q is added to it reversibly at temperature 7\ and £>' taken away from it reversibly at temperature T r we have a T a; -a (32) This property was utilised in defining an additional thermodyna- mical function (Entropy) for the gas which proved to be of great importance in the further development of thermodynamics. In the following deduction, we have followed Swing's treatment.* Let us take a gas in a definite state A (Fig. 1), and compare this state with another state B, which is obtained by the addition of heat energy, Jt is easy to see that the amount of heat added from outside may be anything, owing to the variety of ways in which heat can be added. This can be visualised by drawing curves be- twecn A and B. There may be an in- finite number of curves passing between A and B, corresponding to the infinite number of ways in which heat can be added and for all these the amount &Q of heat added will be different. It will be shown, however, that the integral measured along any reversible process does not depend on the path, but only on the coordinates of A and B. Let us draw two adi aha tics through A and B and draw an isother- mal DC at temperature t cutting these adiabatics in D and C respect- ively. Let us now divide the area A BCD into a number of elemeutai'V Carnot eyries #yyV as in Fig. 1 (xy, x'y r , isothermais, and xx', yf adiabatics). Let SQ be the amount of heat absorbed at temperature T while *Sce Glazebrook, A Dictionary of Applied Physics, Vol. 1, p. 271. --- --vC V Fig. 1. — The entropy function. 252 THERMODYNAMICS [CHAP. *] ENTROPY OF A SYSTEM 253 the point shifts from x to y'and Sq be the amount given up while it shifts from / to x'. Then we have HI 7 t ' Adding up for all the Carnot cycles between A and B, we have M r -4-T* m This is the value of along- the zigzag path Aabxy, B, for the value of the integral along the adiabatics Aa, bx, ....is 0. In the limiting case when the Carnot cycles are infinitely small the zigzag path Aabx B coincides with AB, Hence I If-f=><- (34) r B so The value of die integral j -2?- is now seen to be equal to 'XBq/t and hence it is independent of the path between A and B. Instead of taking the particular isothermal DC we may draw any other isothermal cutting AD and BC, the value of the integral cannot be thereby changed, for if we construct a number of Carnot cycles between the first and the second isothermals we have 1 ' * I ... . - S8 ? = -, E&q*. The points A and 1 ! taken anywhere on the adiabatics AD. and BC respectively, the value of the integral will remain the same, for J ^ along an adiabadc is zero since S£> = Q. Hence it follows that if we pass from one adiabatic to another the expression I — =* J % T denotes a quantity which increases by a fixed amount independent of the manner of transformation. If we start from some zero state and denote the value of the integral f -£- by s A it follows that Jo ■* function of the coordinates of A alone. Clausius called it the entropy of the substance. Now since we are not familiar with the zero state of the body when it will be deprived of all its heat, we cannot, find in this way the absolute value of the entropy in any state. We arc- however generally concerned with changes in entropy and hence we can measure from any arbitrary zero as is done in the case of energy. To measure the entropy m any state we have to take the substance along any reversible process to its zero state and find the value of lAq/t along this process. The entropy, though conveniently meaaur- jj. is a ed with respect to a reversible process, has nothing to do with it and exists quite independently of it. The above treatment is perfectly general and w r ill hold whatever substance is taken as the working substance. Now if the work is done by expanding against an external pressure we have from the first law for any process reversible or irreversible, &Q = du -\- pdv, where p stands for the external pressure at every stage o[ the process. But for finding entropy we have to substitute the value of Sf> along a reversible process. Now the process is reversible only when "the external pressure p is always equal to the intrinsic pressure of the substance, i.e., when the expansion is balanced. Hence we get* J = j r J«+M' m where p denotes the pressure of the substance itself. It is easily seen that entropy is proportional to the mass of the substance taken, for if we take M grams of the substance dU — Mdu t dV =z Mdv and hence 5- Ms. Exercise. — Find the increase of entropy when 10 grams of ice at 0°C. melt and produce water at the same temperature, given that the latent heat of fusion of ice = SO calories/gm. Since the process is reversible and isothermal we have AS = i|£ 3 — =2.93 caL/degree C. 18. Entropy of a System* — In the last section it was shown that if the entropy per unit mass of a substance is s the entropy of m grams of the substance is ms. Tt can be shown in a similar way that if we have a system of bodies in thermodynamical equilibrium with different thermodynamical variables and having masses m 1t nu. . and specific entropies s s , s 2 . . ., the entropy of the whole system "is S = »»!*! + m 2 s 2 + ... . . . . (30) 19, Entropy remains Constant in Reversible Processes.— The sum of the entropies of all systems taking part in a reversible process remains constant. Considering the Carnot cycle we notice that the working substance has the same entropy at the end of the cycle as at the beginning since it returns to the same state. The loss in em of the source is Qi/Ti, the gain by the condenser is Qj/.T* and hence the net gain in the entropy of the system is T B -T L - U < 3 '> Thus the entropy of the whole system remains constant. * A more general definition is the following : — If from any cause whatever the unavailable energy of a system with reference to another system at TV under- goes an increase A£, then A-E/7V measures the increase of entropy of the system. 254 1H KRMODYN AM I CS [chap. In the isothermal expansion of the working substance in the first part of the Carnot cycle the increase in entropy of the working sub- stance is (li/T u the loss by the source is Ql/T\ and hence the total entropy remains constant. This holds for any reversible transforma- tion* Working out as on p. 215 we see that the increase in entropy of a gram-molecule of perfect gas in isothermal expansion from volume F 3 to V 2 at temperature T is but by The change in entropy during a reversible adiabatic compression or expansion is zero. For in such a process the external work done is I pdV and change in entropy is /\S = \ ^ — the condition of the process from the first law dU-\-pdV =0, hence the entropy remains constant in an adiabatic process. Adiabatic curves are therefore sometimes known as "Iscntropics." 20. Entropy Increases in Irreversible Processes. — The entropy of a system increases in all irreversible processes. As examples of such processes we may mention the conduction or radiation of heat, the rushing of a gas into a vacuum, the inter-diffusion of two gases, the opment of heat by friction, flow of electricity in conductors, etc <vt the theorem here for a few cases. Entropy during Conduction or Radiation of Heat, — Suppose a small quantity of heat (? is conducted away from a body A at temperature T, ro another body B at temperature T^{TiyT 2 ) , and I'. so small that the temperatures Tj and T 2 of the bodies arc not appreciably altered in the process, then A loses entropy equal to , r ~ while B gains entropy by an amount J? gain in entropy of the two bodies is Thus the total **=**(-* -it) m which is positive since Ti>T 2 . Hence the entropy of the system as a whole increases during thermal conduction. Similarly in the case of radiation, if a quantity of heat SQ is radiated from the body A to the body B, the gain in entropy is given by the same expression. Thus increase of entropy is always produced by the equalisation of tem- perature. (ii) Gas rustling into a vacuum. — Consider a perfect gas in a vessel rushing into an evacuated vessel and let the whole system be isolated. Since the gas is perfect the temperature does not change * In denotes natural logarithm. X. ENTROPY OF PERFECT GAS 255 in the process. The gain in entropy may be obtained by finding the value of I -=■ along any reversible path connecting the initial and final states. The reversible process most convenient for this purpose is an isothermal expansion of the gas against a pressure which is .always just less than the pressure of the" gas. For a perfect gas dV at constant temperature is zero, gain in entropy A$=4rf pdV = Rln A Hence the (40) where P it F 2 denote the initial and final volumes of the gas. Hence the entropy of the system increases in this irreversible process. 21. Tne Entropy of a Perfect Gas. — Let us take m grams of a perfect gas having die temperature T and occupying the volume v. The entropy ms is given by the relation 7" du-r-pdv o r~~ * Since du = mc„dT, p=zmRT/Mv f we have ms — ms = m (c v In T 4- -rj In v) 4- const. . (41) We can also express wis in terms of pressure. From the relation R M tp - e w -r wet get ms = m(c p In T — -^ In p) -\- const. (42) Exercise. — Calculate the increase in entropy of two grams of oxygen when its temperature is raised from <J D to 100°C and its volume is also doubled,. &S =2x2-3026 5-035 log 373 273 Second + I log 2 =0-184 cah 32 "*" 273 ' 32 ° j deg 22. General Statement of Second Law of Thermodynamics, — 'We have shown above for a few irreversible processes that increase of entropy takes place during such a process. This result is however very general and holds for any process occurring of itself. We shall not, attempt to prove here this general statement, but only enunciate it in the following words : — Every physical or chemical process in nature takes place in such a way that the sum of entropies of all bodies taking part in the process increases. In the limiting case of a rever- sible process the sum of entropies remains constant. This the most general statement of the Second Law of Thermodynamics and is iden- 256 THKRMGDYNAMICS [CHAP. *•] PHYSICAL OONCKPT OF ENTROPY 257 tical with die Principle of Increase of Entr* viz,, the entropy a system of bodies tends to increase in all processes occurring in nature> if we include in the system all bodies affected by the change. Clausius summed up the First Law by saying that the energy of the world remains constant and the Second Law by saying that the entropy of the world tends to a maximum* Though the terras 'energy of the world' and 'entropy of the world' are rather vague, still, if properly interpreted, these two statements sum up the two laws- remarkably well. 23. Supposed Violation of the Second Law.— Maxwell invented an ingenious contrivance which violates the Second Law as enunciated above. Following Boltzmann's idea, he imagined an extraordinary being who could discriminate between the individual molecules. Sup- pose such a creature, usually known as Maxwell's demon, stands at the gate in a partition separating two volumes of the same gas at the same temperature and pressure and, by opening and shutting die gate at the proper moment, allows only the faster moving molecules to enter one enclosure and the slower molecules to enter the other enclosure. The result will be that the gas in one enclosure will be at a higher temperature than in the other and the entropy of the whole system thereby decreases, though no work has been done. This apparently violates the Second Law, We observe, however, that to Maxwell's demon the gas does not appear as a homogeneous mass but as a system compound o[ discrete molecules. The entropy does not hold for individual mole- cules, but. is a statistical law and has no meaning unless we deal with matter in bulk, 24, Entropy and Unavailable Energy.— Consider a source at temperature T-y and suppose it yields a quantity of heat (> to a work- ing substance. If the lowest available temperature for the condenser of the Carnot cycle is T n the amount of heat rejected by the Carnot engine working 'between T t and T is QT /T V The remainder, i.e, Q— Q=? has been converted into work. Thus the available energy i s Q\ * — J- ) ■ Now suppose a quantity of heat $() passes by conduction from a body at. temperature T 2 to another at temperature TV The un- available energy initially was (T iY /T.,)Hl and finally it was (T /r 3 )SQ. Hence the gain in unavailable energy is SQ \Y"" t) T °' ^° bemg the lowest available temperature. The increase in entropy of the system is, as we have already proved, equal to SQ Isr- y). Hence the increase of unavailable energy is equal to the increase of entropy multiplied by the lowest temperature available. Thus entropy is a measure of the unavailability of energy, and the law of increase of entropy implies that the available energy in the world tends to zero., Le<, energy lends to pass from a more available form to a less available form. This is the law of degradation or dissipation of energy mention- ed on p. 243 and is thus seen to be equivalent to the Second Law* It follows as a corollary that all transformations, physical or chemical, involving changes of energy will cease when all the energy of the world is run down to its lowest form. 25. Physical Concept of Entropy,-— The entropy of a substance is a real physical quantity which remains constant when the sub- stance undergoes a reversible adiabatic compression or expansion. It is a definite single-valued function of the Lhermodynaraical coordinates defining the state of the body, viz., the temperature, pressure, volume or internal energy as explained on p. 237. It is difficult to form a tangible conception of entropy because there is nothing physical to represent it ; it cannot be felt like temperature or pressure. It is a J statistical property of the system and is intimately connected with ' the probability of that state. Growth of entropy implies a transition from a more to a less available energy, from a less probable to a more probable state, from an ordered to a less ordered state of affairs. The idea of entropy is necessitated by the existence of irreversible pro- cesses. It is measured in calories per degree or ergs per degree. 26. Entropy-Temperature Diagrams. — We have represented the Carnot cycle and other cycles in Chap. IX by means of the usual in- dicator diagram in which the volume denotes the abscissa and the pressure denotes the ordinate. Another way to represent the cycle, which is often found very useful, is by plotting the temperature of the working substance as ordinate and the enixopy as abscissa. Its im- portance is readily perceived for during any reversible expansion the increase dS in entropy is given by dS = <f, where the integration is performed between the limits of the expansion. Thus the area of the curve on the : 1 1 iropy-leniperature diagram* represents the amount of heat taken up by the substance. The Carnot cycle can now be easily represented on the entropy-temperature diagram by Fig. 2. AR represents the isotlicrmal expansion at T\, BG the reversible adiabatic expansion, CD the isothermal compression at T 2 , the temperature of the sink, and DA the final adiabatic compression. It is evident that lines AB and CD will be * This is also called iephiyram as the symbol <f> was previously used for entropy, 17 D T* 5! I* * Entrtfpy b Fsg._ 2,— Entropy-temperature diagram of Carnot cycle. 258 THERMODYNAMICS [CHAP. parallel to the entropy-axis and BC, AD will be parallel to the tem- perature-axis. The amount of heaL taken in is represented by the area ABba, heat rejected by the area DCbu, the difference ABCD being converted into work. These areas are respectively A (.$' - S), T 2 (& - 5) , (T t - T s ) (S f - S) and the efficiency is 1 - r 2 /7Y 27. Entropy of Steam. — For enabling 1 the student to have a pro- per grasp of the conception of entropy we shall calculate the entropy o£ steam, Let us start with 1 gram of water at, 0°C. and go on adcf. trig heat to it. The increase in entropy when we add a small amount Of heat 8Q at T 6 is -p , But. SQ == vdT where o- is the specific heat ■of water at constant pressure. Hence the entropy of water at T° is , cT ff dT . T if tr is assumed to be constant. s denotes the entropy of water at 9°C) {T (1 D K.) and it is customary to put it equal to" zero. Hence for water - o log, --, This is represented bv llie where o- can he put equal to unity. logarithmic curve OA in Fig, 3. If the liquid is further heated it boils at the temperature T x ab its latent heat L. Since the temperature remains constant in this process the increase in entropy per gram is L/T v This is repre- sented by AB in the figure. Hence the entropy of 1 gram of dry satu- rated steam is If the steam is wet, q being its dryness, i.e., q is the proportion of dry steam in the mixture expressed as a fraction of the whole, the entropy of wet steam is Tf the steam is superheated to T.. c the entropy is further increased by the amount P dT where Entropy Fig. 3. — Entropy-temperature diagram o£ steam. denotes the specific heat of steam during superheating. This is repre- sented by BC in the figure. Hence x.l APPLICATION OF THE TWO LAWS 259 the entropy oL superheated steam at 7V = <? 1°S* n i- + r IS *P dT T As c* v 273 ' T x j Tl ,aries with temperature, its average value over small intervals this way the entropy of steam in any state can be is talten. In calculated. Exercise. — Find the entropy of saturated steam at a pressure of 74 lb. per sq. in. From the table the boiling point at this pressure is 152.G°C. and the latent heat at this temperature is 503.6 cal./gm. The entropy of 1 gmt of water at 152,fi°G or 425.6°K is 1 X log, 425.6/273 = 0.44 c&L/degree. The increase in entropy due to the evaporation at 425!g°K is 503.6/425-6^ 1-18 cal./degree. Therefore the entropy of steam at 425.6°K is given bv .44 -f- 1,18 = 1.62 cal./degree. This may be verified by looking at the steam tables. For more accurate calcu- lation the variation of the specific heat of water with temperature should be taken into account. APPLICATIONS OF THE TWO LAWS OF THERMODYNAMICS 28. General Consideration*.— For making frequent and effective use of the Second Law, it should be expressed in a convenient form. For this purpose a mathematical formulation is more desirable and was supplied by Clausius (p, 252) , was., &Q-r<fe* ( 4 *) where dQ is the heat absorbed during a reversible process connecting the two states. From the first law $Q=du + pdv, (7) where p denotes the intrinsic pressure of the substance and is given by its equation of state. Combining these two equations we get du + pdv=Tds (44) Let us further investigate the nature of the variations du, dv and <fo. As already stated in Chap. TV and also on p. 237, tire state of every substance is fixed if we know any two of the variables p, v, T, U....L For example, the internal energy uz=f(v, T) is uniquely defined it v and T are given. Thus if we represent the internal energy u of the substance" on the coordinate svstem v, T , the value of u for every point is fixed and does not depend upon the path along which we bring the substance to that point (state) ; in other words, the total change du has the same value whether we first make the change dv in v'and then the change dT in T or in the reverse order. * The results deduced from this equation in the following pages i and their confirmation by experiments may be taken as direct verification of this equation and therefore: of the Second ! ,r>\". 260 THERMODYNAMICS CHAl*. Mathematically, this means that du h a perfect differential and as explained on p. 239 we must have O'u a a — V7S (45) BxBy djdx where x and y are the two variables defining the state of the system. The entropy function was shown on p. 252 to be such that its value does not depend upon the path of transformation, i.e, f s is also a single-valued Junction of the coordinates. Similar is the case with the volume and the internal energy. As a contrast and for clearer understanding, it may be mentioned that pdv and lQ-= du -\- pdo are not perfect differentials for their values depend upon the path of transformation. This may be easily seen as T pdv represents th^ area of the indicator diagram and has different values for different paths. But though du -\- pdv is not in le- viable we have proved by physical arguments that (du -f- pdv) /T is in tegrable ; in other words, 1/7* is the integrating factor to the energy equation. We now proceed to make use of the above ideas for the further development of equation (44) . 29. The Thermodynamkal Relationships (Maxwell).— Equation (44) may be written as du = Tds-pdv. .... (46) Now if x and y are any two independent variables we have, since «, s and v are expressible in terms of x and % , Bs Bs , j dv , , do , dv — ■=— ax- ■- ay, Bx By ■" and du = du , . Bit , Substituting the values of ds, dv, du in (46) and equating the co- efficients of dx t dy, we get <?«_ ~Bs _ dv Bx~ Bx ~ ** dx' Bv_ T ds _ dv "By 'dy p dy Now, since du is a perfect differential, we should have d*u _ _ B'ht BxBy~dyBx' B ( r -ds jv\ b 1 . r ds dv\ • X. FIRST RELATION BxBj Expanding dydx out and remembering that we get B*i> B z v dxdy dydx 261 and (4?: er ds _bt B± = &p<h_ bp b^ dx" By By "Bx 3x By By Bx' which when geometrically interpreted means that the corresponding elements of area whether represented on T, s or p, v coordinates are equal. This relation will hold for a simple homogeneous substance. For convenience of remembering we may write (47) as B{T*s) _B\p : b) d(x,y) 3{x,yj ' where '- '--stands for the determinant B{x t y) BT BT Bx dy Bs ds Bx By Any two of the four quantities T, s, p, v can be chosen as x t y. This can be done in six different ways and correspondingly, we have six thermodynainical relationships, though all of them are not independent. 30. First Relation, — Let us take the temperature and the volume as independent variables, and put x = T, y = v in (47). Then BT _ , Bv_ _ . Bx By and $1 By dv P = o, Bx (48) since T and v are independent. Hence we have (*) ={%) ■ ■ ■ ■ \ to/ r \3r } v which implies that the increase of entropy per unit increase of volume at constant temperature is equal to the increase of pressure per unit increase of temperature when the volume is kept constant. We can apply equation (48) to the equilibrium between the two states of the same substance. Multiplying both sides by T to It \dTf v (49) which means that latent heat of isothermal expansion is equal to the product of the absolute temperature and the rate of increase of pressure with temperature at constant volume. Thus, if a body 262 THERMODYNAMICS CHA1\ changes its state at T Q and absorbs the latent heat L, and the specific volume in the first and the second states are v x , v 2 , equation (49) yields L _ dp_ v*~-v. at or dp_ dT (50) This is Clapeyron's equation and is one of the most useful formulae in thermodynamics, A more rigorous proo£ of this will be given in sec. 39. Let. us first employ (50) to find the change in the freezing point of a substance by pressure. In the case of water at o C 1 £, = 79.6 X 4,18 X 1A T ergs per gm. T = 273. 1*2 = 1.000 e.e. (specific volume of w r ater at 0°C), 0!= 1.091 c.c. ( „ „ „ ke at 0°< dp__ 79-6 x4-l 8x10- '* dT 27S x (1-1091)' Now if dp — 1 atmosphere = LOIS X 10 6 dyne/an 2 , 8T= -0.0075°. This shows that the melting point of ice is lowered by increase of pressure, the lowering per atmosphere being 0.0075°. The pressure isary to lower die niching point by PC. is 1/.0075 = 133 cm*. This accounts for the phenomenon of Regulation of ice and the experiment of Tyndall (p. 110). These results were quantitatively verified by Kelvin, His experi- mental results are given in Table L Table 1 .— Depression of freezing point with titer ease of pressure. Increase of pressure. Depression of freezing point in °C. observed. calculated. 8.1 aim. 16.8 „ 0.0580 0.1289 0.0607 0.1260 For substances which contract on solidification the melting point will be increased by pressure. Thus for acetic acid whose melting point is 15.5°C de Visser found experimentally that an increase of pressure by 1 atmosphere increases the melting point bv 0,02432°C < , while equation (50) gave 0.02421 °C. This verifies relation (50), x.] FIRST RELATION 263 Equation (50) will also hold in the case of fusion of metals. With its help the change in melting point of any substance with pressure can be easily calculated. The values so calculated, together with the experimentally observed values, for a few metals are given in Table 2. The agreement is fairly satisfactory. Table 2. — Change in melting point with pressure. Metal. So Cd Pb Bi Melting point in °C. 2 M 1 .0 320.4 326.7 270.7 Latent heat in cal./gm. in c.c. per gm. &T per 1000 atm. (cak.) 1 t.25 13.7 5.37 12.6 0.003894' +3.34 0.00564 | +5.9) 0.0039761 +8.32 -0.00342 -3.56 ST per 1000 atm. (obs.) +3.28 +6.29 +8.03 - &.55 Equation (50) may also be employed to calculate the latent heat. For the vaporization of water at 100°C. we have the following data :-- T — 373°, ?'i = 1 c.c, w 2 = 1674 c.c. per gram. iL_ —27.12 mm. of mercury, a I . 373x(1674~l)x27-12*l-Q13xl0« ^ n ^ f " L ~ 760 x 4-185 xlO 7 '* The accurate experiments of Henning give L = 538.7 cal. per gin. at 100° C. which is very close to the calculated value. We can also calculate the latent heai, of evaporation of water at various tempera- turfs from (50) if we know the values of v 2 and ^ at these tempe- ratures. The values of L so calculated are given in Table 3, together with the observed values. The agreement is seen to be very close. Table 3. — Latent heat of steam at different temperatures. t°C. dp -&=. in mm. 100 110 120 130 140 150 160 170 180 dT of Hg, 27.12 36.10 47.16 60.60 76.67 95.66 117.7 143.4 172.7 v 2 in c.c. 1674 1211 892.6 669.0 509.1 393.1 307.3 243.0 194.3 L (calc.) 539.5 533.4 526.6 520.1 512.9 505.7 497.5 489.8 481.8 L (obs.) Henning 03 .7 .1 525.3 518.2 510.9 503.8 496.6 489.4 482.2 "Taken from Jellinek, Lehrbuch dcr Fhysikalischcn Chetnk, Vol. 2, p. 517 (1930 edition). 264 THERMOD YNA MICS [chap. 31. Application to a liquid film, — Equation (48) may also be applied to the case of a liquid film. If such a film is. stretched, its volume remaining constant, the energy equation for Lhis case is ($Q) s = du-2<rdA, for the work done by the film is -2odA, where <x is the surface tension and dA the increase in area of the Film, The usual term pdv on the right has disappeared since the total volume remains constant. Hence corresponding to p and dv we have in this case -2,q and dA respec- tively. Therefore equation (49) yields \m)t,v = " 2T \dT)»,A and for a finite change •q—it (£),*• (si) For a liquid the surface tension decreases with temperature, therefore — r is negative and 8£> is positive. Hence an amount of heat must be supplied to the film when it is stretched in order that its temperature may remain constant. In an adiabatic stretching the temperature will fall by an amount «■- ££(£■.) dA (52) where C A is the heat capacity of the film. 32. Second Relation, — Another important application of the thermodynamic formulae consists in their application to some adia- batic changes such as the sudden compression of a liquid or sudden stretching of a rod. For this case let. us put x = T, y = p. Equation (47) then reduces to &)r — (£).• • • • ™ which means that the decrease of entropy per unit increase of pressure during- an isothermal transformation is equal to the increase of volume per unit increase of temperature during an isobaric process. Multiplying both sides by T we have {£),=- r (ia=- r - ■ • < m > where a is the coefficient of volume expansion at constant pressure. From tins relation it follows that if a is positive, i.e., the substance expands on heating, | jM is negative, and hence in this case an amount of heat must be Laken away from the substance when the *■] SECOND RELATION 265 pressure is increased, in order that the temperature may remain constant. That is, heat is generated when a substance which expands on heating is compressed. For substances which contract on heating,, a cooling should take place. These conclusions were verified experimentally by Joule who worked with fish-oil and water. The liquid was contained in a vessel closed at the top by a piston and pressure was suddenly increased by placing weights upon the piston. The change in temperature was measured by a thermopile. From (54) we can deduce the increase in temperature AT produced by a sudden increase of pressure A/>- We have* PB» where c$ is the specific heat of the substance and v its specific volume. We have assumed that v a /c p is independent of pressure, and then integrated for a iinite change /\,p in p. Joule's results with water are very interesting and are given in Table 4. Table 4. — Increase i?j temperature of water by sudden increase of pressure* Ap in kg. per cm 2 Initial temp. in °C, AT (obs.) AT (calc) 26.19 - 0.0083 - 0.0071 26 J 9 5.00 +0.0044 +0.0027 26,19 11.69 0.0205 0.0197 26.19 18.38 0.0314 0.0340 26,19 30,00 0.0541 0.0563 26.17 31.17 0.0394 0.0353 26.17 40.40 0.0450 0.04 7 U The agreement between the observed and the calculated values is seen to be very close. This proves the essential correctness of the theory ; in fact, these results formed one of die earliest experimental verifications of the second Jaw. Thus the thermodynamic theory explains the remarkable fact that water below 4°C. cools by adiabatic compression inspite of the fact that the internal energy is increased. Another series of experiments consists in the adiabatic stretching of wires. The best results were obtained by Haga. The change in temperature of the stretched wire was measured by means of a thermopile formed by the wire itself and another thin wire wound round it. It will be seen that tension means a negative pressure and hence wires of substances which expand on heating should show a cooling when stretched adiabatically. In this case the work done *We may directly deduce this result from equation 266 'III liRMOD YN AM ICS dv iCtJAP. is not pdv but -JPctt hence ?=. must be replaced by ^ where I is the length of the wire. Hence in place of (55) we get where /3 is the coefficient of linear expansion, ta the mass per unit length of the wire, c p its specific heat in mechanical units and C the heat capacity of the wire. For a German silver wire of diameter 0.105 cm. at room tempera- ture, Haga found the mean value AT = -0,1063 for a tension of 13.05 kg. and AT = -0.1725 for 21.13 kg. tension, giving a mean value of 0.00813 per kg. If we substitute the values o£ w, c p , /3 in equation (56) and use the value / — 4.1S X 10 7 ergs per tab, we get AT =-0.00810 in dose agreement with the experimental value. India-rubber when moderately stretched has a negative expan- sion coefficient. This should show a heating effect when further stretched adiabatieally, which is found to be true experimentally and may even be felt by the lips. 33, Other Relations.— Besides the above two, there are other rtant relations. Thus putting x — S* y = v in (47) we act IS).-- (ft- • • ■ ^ Again putting X ss s, y =/j we get This is the third i 9 —) = (— ) (58) This is the fourth relation. The interpretation of these results is left to the reader. The above relations are known as Maxwell's four thermodynamic relations. Besides these, there are two more relations which may be obtained by taking p, v t Or T, j, as the pair of independent variables. They are GH.Or-eirew.- 1 - • ■ « These arc called the fifth and sixth relations and are much less used. 34. Variation of Intrinsic Energy with Volume.— From the tela- tta(£) r -(§L) p . we find by substituting ^±^' for ch x.] that VARIATION OF INTRINSIC ENERGY WITH VOLUME ©rT*{fr).-* 267 (61) This equation enables us to calculate the variation of intrinsic energy with volume. For perfect, gases* p = y, hence |^J ^ = (Joule's law, p. 47). BT For gases obeying the van der Waals' law p = ,^» — y\ it can be easily seen by substituting in (61) that For general systems, we can transform (61) to a form which allows du us to calculate J- from experimental results. This follows from do (68) equation (IB) of page 240 where it is shown that \dTt,r \BvlT\dTlp = TEa-Pi ..... (64) where £ = bu1k modulus of elasticity, « = coefficient of volume expansion. We shall compare the ratio -. u : p for a typical liquid, viz., mercury. For mercury at 0°C and atmospheric pressure we have as on p. 240. (dP) - 46.5 atm. em-* ^C- 1 1(|"\ = 46.5 X 273.2 - 1 = 12700. Hence du Now — is sometimes known as the internal pressure the idea being that the visible pressure is equal to the force per sq. cm, with which the particles bombard a layer inside the liquid minus the internal pressure, i.e. force/cm 2 with which they are drawn inwards due to forces of cohesion. The above comparison shows that for * U, V refer to a gram-molecule, hence ay = ~qZ m £ e » el 'al. hi the text both forms have been used. TH ERMODYN AMICS CHAP. ordinary liquids, this force is very great but when we increase the temperature, the ratio diminishes till it vanishes when the perfect gas stage is reached. 35* The Joule-Thomson Effect. — In Chapter VI, we proved that in the Joule-Thomson expansion u -J- pv remains constant while the pressure changes on the two sides of the throttle valve. Let us calculate the change in temperature due to an infinitesimal change in pressure during this process. We have by the conditions of the process d (u -f pv) — (65) Since from the two laws of thermodynamics du -\- pdv ss Tds, we have from (65) Tds + vdp = Q, .... (66) or \ QW Hence or r (a 8 V)/ T+T (|) r * +t * = °- T \dT) p ==cp; [dpfr^ " XdTlp rlBv \ \dp )h c„ — o (67) - — | . op Ik For a perfect gas, TK J - V is easily seen to be zero and the Joule-Thomson effect vanishes. The porous plug experiment therefore provides a very decisive test of finding whether a gas is perfect or not. For a gas obeying van der Waals* law it can be proved that 1B_T\ = 2a{V-b)*-bV*RT V [dp (B_T\ = 2t W hi = 1 RTV*-Za{V— bf C; -idr-*) 1 **" JVOX, (68) ST Hence ^— is positive, Le. when &p is negative there is cooling as long as 2*<t„- . At T greater than ZafbR the gas becomes heated on suffering Joule-Thomson expansion. The temperature stands for the total heat u + pv or the enthalpy of the system. x.] CORRECTION OF GAS THERMOMETER 269 given by T. — 2a/bR is called the temperature of inversion since on passing this temperature the Joule-Thomson effect changes its sign. It will be easily seen that —=—T s approximately where T e is the critical temperature of the gas. We have already dwelt upon this point in Chap. IV, These results have been approximately verified by experiments, 36. We will analyse the Joule-Thomson effect in another • which perhaps gives a better insight into the mechanism of the phenomena. We have Now Tds = du 4- pdv, . , l*Z\ - _/*l\ ( S (P»)\ " M3> /*" Wt \ dp IT (69) The first term on the right, measures deviation from Joule's law while the second gives the deviation from Boyle's law. and the Joule-Thomson effect is the resultant effect of deviations from both these laws. Now {-'-) is always negative, for that part of the internal energy which is due to molecular attractions always decreases with decrease of volume, Le., increase of pressure. Hence due to devia- tions from Joule's law alone the Joule-Thomson effect will be a cooling effect. Upon this will be superposed the effect of deviations from Boyle's law which is a cooling effect, if d(pv) dp is negative (t.«r v &f» before the bend in Fig, 6, p. 98) and a heating effect if —fr' is positive. 37. Correction of Gas Thermometer, — Equation (67) can be directly utilised for giving the absolute thermodynamic scale from observations on an ordinary gas thermometer. For we have 'fr).-*GV *P where all these quantities ought to be measured on the thermo- dynamic scale. In actual practice we use a gas or any other thermo- meter to measure this temperature which we may denote by 6 and c/ denotes the specific heat on this arbitrary scale. We have _ dO _ dCL d9_ , </0 6 *~~ df d& dT cp dT' 270 Further TIIERMODYN AM ICS ldT\ _dT tm \dplk d$ \dpi h tar/," \dBfpdT' [CHAP. Hence (67) yields dT T (!) v+c. \Bpn or (70) The quantities occurring on Lhe right-hand side can be measured on any thermometer ; all that is necessary is that the same thermometer should be used in all these measurements. Further these quantities vary with temperature. As a first approximation* let us assume them to be constant. Then on integration we get log T = log (v + pc/) + const. dp h constant. This shows that the thermodynamic temperature T is related to the volume of the gas in this complicated way and not only as T qc v as in the case of perfect gases. Let the thermodynamic temperatures corresponding to melting point of ice and boiling point and Tioo respectively and the corresponding volumes of the gas he u and v tm . Then ,, stands fior the Joule-Thomson effect r«— ) and a is a T"n ^ 1IM ' 100 'j ; :'-'"; be the volume coefficient of expansion. Then r.-- L(H-fi«l). (71) 1 Ins gives the temperature of the melting point, of ice on the thermo- dynamic scale. The results for a few gases have been calculated and are given in Table 5, For hydrogen the Joule-Thomson cooling is -0.039°C. per atm., Me t = 6.86 X 4.18 X 10 7 ergs per mole per °C, Mv = 22.4 X 108 c, c , per mole and a == .003661 S per D C. * More accurately, however, we have to integrate over short intervals. *■] .'. T = 273-13 ( I — ' EXAMPLES 039x6*86x4-18x10 271 ! ) 10 e x 22-4x1 275.13 (1 -.00050) =273,0 degrees. Table 5 Joule-Thomson correction to the gas thermometer. Uncorrected Volume Mean Joule- temperature Correction Thermo- Gas. coefficient Thomson of melting ice term dynamic of expan- cooling per fl 1 e p ' se tempera- sion a atm. #o = - 0% dp ture T per °C. H-, .0036613 - 0.039°C. 273 J 5 -0.13 273.0 .0036706 +0.208 272.44 +0.70 273,14 co 2 .0037100 +1.005 269.5 +4.44 273.9 For air the dnin are most reliable and yield the value To = 273.14°. It will be seen that though for the various gases the melting- point on the uncorrected gas thermometer (0 O = l/« in column 4) is much different, the melting point corrected by making use of the Joule- Thomson effect comes to about the same value, viz., 273°, To find the correction to be applied to the gas thermometer at other temperatures we must employ (70). Thus comparing ^ (70) and (71), we have a .= l/v a. Then any temperature T t is easily found if v., measure the corresponding specific volume and the Joule- Thomson effect ; for r,-+(* +f ,de\ (72) The eas thermometer temperature is 9i s= — - — , and the correction ° v a a . ' -aa term is — *- - r . Rut as already mentioned on paste 10 the existing v a a dp l * data on Joule-Thomson effect are not sufficient, and the corrections are usually calculated from deviation from Boyle's law. 38. Examples. — We will now give a few examples which will illustrate the utility of the foregoing thermodynamic formulae and will give some practice in applying them, 1. Prove that c p - c v = ^(^) (|y) = TE#v, where E h the bulk modulus of elasticity, a the coefficient of volume expansion and v the specific: volume. 272 THERMODYNAMICS CHAP, *■] :- 'I I CIFIC HEAT OF SATURATED VAPOUR 273 ["»-+-rfr) r Tb) m+ T%) r l») t r feh— fc)r-(fr). ••• "— (Ir).(fH [ We have |^J ^j ^) , Differentiating partially with respect to T we get ^ = ^ , Now * p = 7" ( | J f ) and therefore U» It ' Bv5T §Tdv ~ l tar*/ J 3. Prove that Stan From (bp)t = ' (bt) p and P rocecd * above - ] 39. Clapeyron's deduction of the Claasius-CIapeyron Relation.— equation was deduced on p. 262 but the method emploved there is open to objection because Lhe thermodynamic relations hold rigor- ously for a homogeneous substance and 'their extrapolation to dis- continuous changes is open to question. As however the relation is important, a more rigorous deduction is given below which is due to Ciapeyroii. Let ABCD, EFGH (Fig. 4) represent two consecutive isotherm ah at tempera- tures T and T + dT. From F and G draw adiabatks meeting the second isothermal at M and N. We can suppose a unit mass of the substance to be taken through the reversible Carnot cycle FGNM, for Instance, allowing it to expand iso thermally along FG, adiaba tic- ally along GN and compressing it along NM isothennally and then adiabatically along MF. The substance at F is in the liquid state and at G in the form Fiff. 4,- Clapeyron's deduction ui tlit Clausius-Qapeyron relation. of vapour. The amount of heat taken during the cycle is therefore L 4- dL at temperature T + dT. Therefose the work done during the cycle is from equation (10), p, 216, given by f T . m/ T+dl-T\ r , _ ■ : uiswiixce ueiween e\r anu jvii\, t,e. f tip tne increase u i pressure to increase in temperature by dT, change in volume due to evaporation or I grain of liquid and therefore equal to ya-Vj where v % , tr a denote the specific volumes of the vapour and ;>!: liquid respectively. Hence the area of the cycle is Equating the two expressions for area we get {V % -V t )dp~ i^dT, or (50) _ L Equation (50) h called the First Latent Heat equation or the late; ion of Clapeyron. 40. Specific Heat of Saturated Vapour.— -A simple expression for the specific heal of saturated vapour may be. deduced with the help of Fig, 3. Consider the cycle represented by the curve BFGC. The amount of heat taken by the substance during its passage from Pa to F is c tj dT and that during the pal s L -{- dL. On the other hand Che substance gives back heat equal to c dT during the path GC and equal to L during the path CB. c t denotes the specific heat of the liquid in contact with vapour, and e r the s; of the vapour in contact with liquid (specific heat of saturated vapour). The total amount of heaE taken during th c; BFG i c h dT 4- L -f dL - c s dT - /., and this must, in the limit be equal to the area FGNM, which is UT equal to T as proved in the last sei ■ T+L. ^- % JT-L-- or = c, ~e. (73) L dT ~ r This is called the latent heat equation of Clausius or the Second Latent Heat, equation. 18 HMOUYNAMlCS It may be noted that c s% is neither the specific heat at constant p. inc heat at constant volume. Here the liquid and die vapc rays remain in contact and therefore the vapour always remains saturated, Both the pressure and the volume : | that the condition of saturation is always satisfied, It is easily seen that c s does not appreciably differ from c^ the specific heat at co pressure, for the effect of pressure is too small bo brine about any considerable change in the state of the liquid. c tj ea therefore be put equal to c H . We can now calculate the value of i from equation (73) . For water at 100°C. = -0-64 cal. gmr^C" 1 ; L = 539 cal. gm." a T =-- 373* : c. = 1-01 cal gmr^G" 1 can c t = 101 s 0.64-— = -1*07 caLgm^C-' Thus the specific heat of saturated water vapour at !00 fl C. comes out to be a negative quantity. This is rather radoxical Tesult ame lime perfectly true. In Chapter II we have seen ic heat may vary from + * to - oo depending entirely In the present case the conditio satisfied. i, at 10O°C, and 787.5 mm. at 101 *C. the sp ' ater vapour sated to 101°C. at constant pressure it ■ the condition of saturation has to be be compressed till the pressure becomes 787.6 rhis compression generates heat, and in the case of water at 100*C. the heat generated is so great that some of it must he withdrawn in order that the temperature may not rise 101*. The net result in this case is that heat must be withdrawn from and not added to the system during the whole operation. Thus we expla why the specific heat of saturated vapour sometimes becomes negative. ' The same idea can be expressed mathematically. We have refers to the condition oE saturation being satisfied. Or e t = e. T r , v MM \ST) t \dTl m . x. THE TRIPLK PC 275 Now for all vapours is positive and hence c, is always less than Cp and may even become negative. From these considerations it will be seen that saturated steam must become superheated by adiabatic compression, e.g,, water vapour at 100°C. and 760 mm. pressure., when compressed suddenly to 787 mm. would be heated by 2. PC, and hence become superheated; in other words, when the temperature of saturated steam is raised it gives out heat. Conversely, when it is allowed to expand adiaba- tically, say, from 787 mm. to 760 mm., the temperature would fall by 2.1 "C.j Le., to 98.9°, and hence it would be supercooled, and partial condensation may occur. For certain vapours such as saturated ether vapour, the work done in compressing the substance is not so great and the specific heat is positive. These do not become super- heated by adiabatic compression. These conclusions were experimentally verified by Him in 1862. He allowed steam from boilers at a pressure of five atmospheres (temperature about 152°C) to enter a long copper cylinder fitted witii glass plates at its ends. When all the - air and ' condensed water had been driven out, and the cylinder had attained the temperature of the steam, the taps in the supply and the exit tubes were clos and the vapour when viewed from the ends looked quite transparent. The egii tube -was next suddenly opened and the vapour expanded adiabatically and a dense cloud was observed inside the cylinder. The cloud however soon disappears as the cooled vapour rapidly absorbs heat from the walls of the cylinder which are at 152°C. No such condensation was observed in experiments with ether vapour. 41. The Triple Point— If we now plot the saturation value of p against T we get a curve OA (Fig. 5) the slope at every point of which will be given by £, dT T{v s -v t ) » where L t is the latent heat of vaporisation, and v g} v) denote the specific volumes at the temperature considered. To fix our ideas let us consider the case of water. When the temperature is re- duced to 0°C. water freezes and we get ice. But ice has also a definite vapour pres- sure which has been measured. The vapour pressure curve of ice may be represented by the line OB., the slope at every point of which will be given by dp L, dT T(v t -v,) where L s is the latent heat of sublimation. Temperature Fig. &— The triple point 276 THLRMODYNAMICS [CHAI\ x-1 ii ie TKirLi': PO] 277 Similarly lor the phenomenon of melting, the curve OC repre- sents the relation between pressure and temperature, and the slope at any point is given by \ dT T(v,-v t ) ' where L s is the latent heat of fusion. We have already seen on p, 262 that for ice -?• — 133 X 10° dynes cm. - °C-\ i.e., the curve 1 a r should be almost, vertical. For substances which contract on solidification the slope of OC will be positive. The three lines OA, OB, OC are respectively called vaporization line, sublimation tin •, and in the parti- cular case of water they are called steam line* hoar-frost line and ice line respectively. Consider the substance in the state represented by any point Q above the line OA. It will be noticed that tile pressure of the subs- tance In this state is greater than that which will correspond to the saturated vapour pressure at that temperature and which is given by the intersection of OA with the ordinal a from Q. At this iressure represented by O, therefore, die substance can not boil at this temperature as boiling point is raised by pressure, and mi' lore exist as liquid. Thus the region above OA xe- pres, Similarly for points below OA the correspond- too low and the substance must exist as gas. ■ substance r • as solid and below it any point in OC due to the pressure being Larg g to the ice line, ice will melt and therefore presents water, and that below represents ice. It can be easily shown that these three curves must meet in a single point which is called the triple point. For, if the curves do not meet at a point, let them intersect each other forming a small triangle ARC (Fig. 6) - Then since the space ABC is above AB it must represent the liquid region, but it. is below AC and must therefore repre- sent gas, and is also below BC and must therefore also represent solid. Thiis the re- gion ARC must simultaneously represent the solid, liquid and gaseous states. Evidently it is impossible to satisfy these mutually contradictory conditions and the only conclusion to be drawn is that such a region doesnot exist. In other words, all the three curves meet in a single point. The coordinates of the triple point can be easily calculated from the consideration that at this point, the vapour pressure of water jtjjt. 6.— Impossible intersections of ice, steam and hoarfrost lines. is equal to the melting pressure of ice. The vapour pressure of water at 0°C is 4.58 mm. and at 1 C G, 4.92 mm. Thus the vapour ure rises by 0.34 mm. per degree, and therefore, if I is the triple point the vapour pressure p at the triple point is given by p = 4.58 -f 0.34 t (74) The melting pressure of ice at 0°C is 760 mm. and the change of melting point with pressure is 0.0075°C per atm. Therefore the melting point t at the pres.su re p will be t =t 0.O075 - 0.0075 ~76CT P- (75) Solving (74) and (75) we get t = 0.007455 C C and £ = 4.5824 mm, It will be apparent that since the change in vapour pressure is too small in comparison to the change in melting pressure, it is useless to go through the complicated calculation given above and we can simply assume that p = 4.58 mm. approximately (as t will be very near 0°C) and calculate the corresponding melting- temperature. Thus ( = 0.0075- :0.007455»C. The coordinates of the triple point are therefore t — .0075°C, p mm. At this point three phases (solid, liquid and vapour) co-exist. It was former J y supposed that the curves OA, OB are continuous. first proved by Kirch holt that this is not so. for according to (50) OA at 0°G =■ £ far dT ' ~ T{v t -v^ 607 x4 1 8 i :• .760 — 273x21 xl0*xl0" • = 0.337 ram. per degree. dp , ^ , «i • \ 687x4.18x10?;: ^ for OB (subhmatipn) = ~ m ^ ^ ^ w — 0.376 mm. per degree. The dotted curve OA' is merely the continuation of OA, It represents the vapour pressure of supercooled liquid. At - 1°C\ we have vapour pressure of liquid - vapour pressure of .solid = .04 mm. of mercury. This has been verified by the expe tits of Hoi born, Scheel and Henning. Boohs Recommended. 1. Fermi., Thermodynamics. 2. Planck, Treatise on Thermndynam : - H. Hoaie, Thermodynamics. 4. Smith, The Physical Principles of Thermodynamics (1952), Chapman & Hall. 5. Epstein, Text-Book of Thermodynamics, John Wiley. CHAPTER XI RADIATION 1. Introduction. — Even when a heated body is placed in vacuum, it loses heat In tin is case no heat can lie lost by conduction or convection since matter, which is absolute bote processes, is absent. In such cases v that heat is lost by 'Radiation.' To differentiate this process from conduction, it is enough to note that copper and v lich are ho much different in their conducting powers, cut off radiation equally well when plated between the hot body and die observer. Now heat has been shown to be a form of energy, and die pro- pagation of heat by radiation consists merely in a transference of energy. But the radiant energy in the processes of transference does not make itself evident unless it falls on matter. When it falls on matter and is absorbed, it is converted to heat and can be thereby detected. Let us now study some of the p les of radiant For this purpose we require a soxin it as radiator and some h ment to measure the emitted radiation. As emitter Leslie employed a hollow i cube filled with hot water, whose sides coul- i with different substances. The cul" tnade that it can be rotated about a vertical axis. Such a cube called a] uhe and i in Fig. 1, For die radiation expi nployed a differ* thermometer or a thermopile which are described in the next section. 2. Some Simple Instruments for Measuring Radiation. — Leslie, one of the earliest workers, •■'ir-thermmneter which is now only of historical interest. This consists of two equal bulbs A and B (Fig. 2) ing air and communicating with each other through a narrow tube bent twice at right a and containing some non-volatile liquid like uric acid. When the bulbs are at the same temperature the liquid stands at the same at in both the columns C and D, but if one of the bulbs level of the liquid i in the column C and rises in the column D, By noting- the difference in level of the two columns the difference in temperature of the bulbs can he easily calculated. Melloni was the first to introduce thermo- piles for the measurement of radiant energy. In its original form. Fir. 2.— Leslie dif fitlal air tlicr- rn.:inn ter, XI.] PROPERTIES OF RMUANT ENERGY 279 3,— principle of the ale, •ofa thei consisted of a number of bismuth and antimony Is jmned> as indicated m Fig. 3. The left face forms the hot junction, and the right the cold junction. The antii bars and the bismuth bars ihe current flot iio t juncti o n from bismu th to an u rmopile a 1" i : are arranged in the form of a cube such that he hot junctions are at one face and the cold junctions at the opposite face. Such a cube is shown in Fig. 4. The near lace iction and is cos ck. 'lire thick aote the ;„,,; tti attl, generally mica, the differeiii layers, and the thin lines denote the : junctions. The current across the .soldered junc- itimony bar, liters the next layer at B on . Finally the cur, m opi] thermoj inted for use is shown in Fig. 5. is provided with a com rotect from stray radiatio n..r] ,i .. : | fitallic I the hot junction whi in use, Other ' cribed in sections 21-24. 3. Properties and Nature of Radiant Energy.— With the aid bl these sii apparatus, it can be demonstrated that Radia and Li#i ' , and h;' ■ >perticB. We mention he points of res blance. (i) Radium energy, cuum, for we arc i a hot eh < . : lam] ' fi - is highly : I I current passe; through the fi lament which is not -'- to li glow, the thermal radiation coming from tl r can be dete nsiti+e thermopile. lion, like Is in straight lines, — This can be easily for the heat coming from a flame can be easil) ci oi mterpoi Fig-. • $r— The thermopile (mounted). 280 RADIATTO-V [CHAF. _.. i-U Fig, tf.— Verification of : a screen just sufficient to prevent the light coining from ii. On replac- lie flame by a hot noi :t, no heating effect can be detected by a thermopile. The geometrical shape oJ can be verified by cutting out a cross from the screen and holding close to it a piece of wood coated with paraffin wax. The wax will melt in the shape of a cross. (Hi) Radiation travels with the velocity of light* This follows from the observation that the obstruction of the radiation from the sun during a total solar eclipse is immediately accompanied by a fall of temperature. (iv) Radiant energy follows the law oj inverse squat; ,,, tight, — This may be experimentally proved by the simple arrangement shown iu Fig, 6, A is a vessel containing hot water and having one surface B plane and coated with lamp-black, S is a pile. Tt will be found that if we move the thermopile to another position S' (say double Ehe pre ius distance) , the del lection of the galvanomeb :d to it will be unchanged area of die surface of B from which radiation can reach the d four time amount of radiation intensity of radiation un- reduced' to ii; can be easily g two parabolic mirror and B at some disl once apart with their axes in the same line (Fig. 7) . A luminous source placed at the i produces an image at the ! of B. If we replace the luminous source by a hot non-luminous one, say a Leslie cube, and put a piece of tinder at the other focus, the tinder is ignited. This shows that heat and light travel along the same path and hence obey identical, laws. In fact all the laws ""and results of geometrical • to it, Fig. 7. — Verification of laws of reflection for radia XI. OF RADIANT ENERGY AND LIGHT 281 ■') Radiant enerp rum may be obtained by retraction through a prism (sec. 29), (-(■ii) Thermal radiation exhibits the phenomena of interference and d: ■■■'. — Diffraction can be easily observed with the he a concave grating or by employing a ti ission grating made of a number of equidistant parallel wi (vii radiation can also be polarised in the same way as light by transmission through a tourmaline plate. This can be . by the following experiment : — Arrange that Kght from an in- candescent lamp traverses two similar plates or tourmaline. On rotating one of the plates the image will be found to vanish. Replace the lamp by a Leslie cube. Now a thermopile placed in the position occupied by the image will record no deflection ; if however one of the plates be rotated a deflection will be observed. 4. Identify of Radiant Energy and Light — Continuity of Spectrum. — All these observations lead us to the conclusion that radiant energy is identical with light. As we shall see presently, Radiation or Radiant Energy is the general and more expressive term, rather visible light.) being only a kind of radiant energy which has the distinctive power of affecting the retina of the human eye, and thus producing the sensation of colour. But like other kinds of radiant energy light is also converted to heat when it is absorbed by -mi The identity of light and radiant energy can further be seen from the ft >1 ' owi 1 1 g v. Kpe rimen ts : — ■ When we prod- 1 ctrum of the sun by means of a prism T it ts terminated on one side by the red, and on the other side by the violet. But it can be easily seen that these limits are only apparent, and are due to the fact that the human eye is a very imperfect instrument for the detection of radiation, W. Herschel* n'lSOO) placed a blackened thermometer bulb in the invisible beyond the red. and found that the thermometer recorded a rise in temperature. The rise in temperature was observed also when the mometer was placed in the visible legion. Thus W. Herschel discovered the infra-red part of the solar spectrum, and showed that it was continuous, with the visible spectrum. The source of radiation in this case is the sun, which may be regarded as an intensely hot body. But it may be substituted hj ••uffidently hot substance, say a piece of burning coal ; the posi- tive crater of the arc, or a glowing platinum wire, Only in th ■ the intensity of radiation is not great, mid the spectrum does not extend so far towards the violet. Hi) n of light by -.-~A very simple experiment suffice to bring out the point which has been just mentioned. Suppose we take a piece of blackened platinum wire and pass through n Herachel (1738—1822) was bom at Hanover but settled in England. I worked as a musician and later became ski astronomer. He discovered • planet Uranus in 1781 and later discovered! the infra-red radiations. 282 RADIATION [CHAP. XI. EI.ECTROMAGNETIC WA it a continuously increasing current. The function of the current is simply to heat the wire. We find that it becomes warm, and sends out radiant energy, If a thermopile be held near it, t*M nometer connected to it will show a deflection* When a slightly stronger current is passed the wire begins to glow with a dull red light. This shows that the wire is just emitting red radiation o£ sufficient intensity to affect the human eye. Accurate observations! show that this takes place at about 525 ,J C (Draper point) . With asing temperature, the colour of the emitted radiation changes from dull red to cherry red (900 D C), to orange red (1100°C), to yellow (125G*C), until at about I600°C it * es white. Thus the temperature of a luminous body can be roughly estimated visually from its colour. L Such a colour scale of temperature is given in the chart on pp. 25-26, This shows that si waves are emitted by a d body in sufficient intensity only with ini Vice versa, we may argue that when the temperature of the wire is below the Draper point, it is emitting longer waves than the red, but these waves can be detected only by their heating effect. Radiant Energy or Radiation is thus a more general name for Light. It can he of any wave-length from to co as illustrated ie chart i s 283-284. light forms less than 8/4 ths of an ■ " Ex m of the Chart. — We here the mic 1 along the vertical line. Thus (-5) on he wavelength is 10- B cm,, (—2) inda hart visible light extends from ;l4x lHon. (violet). The infr; W. Hei .tension towards the tb side is chiefly due to die reststrahlen method of Rubens and a are detected chiefly b thermopile, for or plates are not sensitive beyond the green. romatic plates may be used, but they are also i p. mi beyond 8000 A. U. Beyond this plates dipped in or neocyanine may be used up to 10000 A. 17. Recently photographic plates sensitive up to 20000 A. U. have been placed on the market. The Hertzian waves are produced fay purely electromagnetic ids (due to oscillation of current in an indue ►acity cir- cuit) and were first discovered by Hertz in 1887. Marconi ap i for wireless transmission (1894). Waves used for this put are generally 10 — 500 metres in length, though waves of about. 10 length have been used in the past. These long electromagnetic from the light waves only in their wavelength, and i :4s have been made to get shorter and shorter waves by purely electromagnetic methods so that one can pass continuously to the f According to a recent experiment by Bs . persons who have r their eyes in the dark carefully, can detect radiation even from a h to about 400 °C. ELECTROMAGNETIC WAVES Wavelength Log Radiation Generation Detection anc Analysis Discoverer cms Siegbahn unit -11 Hard Y rays Reactions Paii- | production effect -10 ^ •0GA.U- '". y-rays > Radioactive Disintegration Photo -electric ■:,hcd 10-*' Angstrom unit ] X-rays. Hard ~ 8 ' Medium Bomhardment targets by electrons Cry* Photography charal jttiintgenue&J Laue mm -7 / Soft i i Compton[i9£5! ThitmudiUtefl 136 A,U. • " 6 *n 1-0XKP-5 •jL t ^ k Vacuum Spark Spark. Arc KLame Photogr aphy . Pri: ma . a : •:: ■.-- ._: 9 - 1 IS ~ .'- 11 8 Millikan ctsi9> & Lyman ami Schumann ■::■:. J: RitterdMfij NEWTON HerSchelO&iO) -3 1 Infra-red Heating Wire gratings & Tharroo- cooples 1 mm - -2' Residual ray Method Rubens & Nicholsassa) '4 mm ■ 4mm- J -1 0. J Hertzian J waves 1 SI Dl Spark gap Coherer Wood ana Nichols & Tears (wao BoBeoffla J mm 284 RADIATION CHAP. XI.] FUNDAMENTAL RADIATION l'UG< ELECTROMAGNETIC WAVES *»w Wavelength b. Radiation Generation Detection and Analysis Discoveror cm s-r Spark gap discharge +i Hertzian waves Triode valve Radar (Micro- 1 metre +-2 Oscillator wave Technique) Hertz 1\m 3 Shert'Wive Wireless Methods 100 metres 4 Herteian 10 «M B WH 7 Ne upper limit B reached are due to between lone infra-red rays. The: limits Lebedew, Lampa Sir J. C. Bose (4 mm.) and the dect^magneti/ waves and beat waves has been completely fan* hv the Nichols and Tear, Arcadiewa, and others. Duiii^ World War II the technique ol prodm tion of waves of lengtn ironi a few centimetres lias undergone a . te revolution . . ,>f magnetron and klystron tubes used toi Rat The ultraviolet part (rays shorter than the last visible = 3800) was d i sc IT. Ritter in 1802. lie round I .notogr aphic plate was affected even beyond the visible limit. < f*« usually give lines as far as A = 3400, after which either uvioi glass flimit A = 2800) or quartz prisms should be used, At about A — <-> the gelatine used in the photographic plates begins to abs , Schumann was the first to prepare plates without gelatine and open up what is called the Schumann region. Below A== I860 A. U. qua begins to absorb heavily, and Schumann therefore used fluontr- Bpectroscopea. Beyond A = 1400, air absorbs heavily, and the whole ope. ! producing light and photographing the spectrum must be done in he pioneer in this field was 1 h Lyman o Han i W hc ; le to photograph Hues as far as A = 600. In this region -, ra1 ; i be used for producing the spectrum. X- f ; shown by M. Lane in 1912 to be light waves of remely short wavelength 10— « cm., w, about 1000 times shorter than ,-, light. He used a crystal as diffraction grating, lhe shortest. X-rays measured by the crystal method has the wave-length I U. -/-rays, obtained in radioactive disintegration of the nuclei oL atoms, are still shorter, viz., from HH> to 10-«> cm, in wavelength. The gap between the Lyman region (600 A. U.) and the soft X-ray region (aboul 20 A. U.) lias been gradually by the works of Millikan and Bowen, Compton, Thibaud and others. Mlllikan and Rowen used ordinary vacuum spectrographs with diffraction g Their source of light was condensed vacuum spark. For ' wavelet i in this region, crystal gratings have then- spaces too small, while tfo ruled gratings have their spaces too large. Conipton instituted method of obtaining spectra in this region with the aid o grating at glancing angle (see M. X. Saba and X. K. Saha, Treatise .Modern Physics, Vol. I, p. 273). Since radiant heat and light are identical, all the laws and theorems of Optics and Spectroscopy can be applied i udy of m. Rut in this chapter, we shall deal with the subject only as far as it is connected with heat. We shall first enter into a preliminary discussion regarding the passage of radiation through matter, 5, FnndamentaJ Radiation Procewe*.— Every hot body emits r; tion from its surface which depends upon the nature of the surJ its size and its temperature. This is known as emisai When the emitted radiation falls on matter, a part i\ reflet 286 RADIATION [CHAP. and transmission are connected by the relation absorption where r — fraction of total energy reflected a — " " » » absorbed i . _ — M « » « transmitted. all ^ 'f CB n ° r t S nsm ^ the % ht ««<* falf/onTTC ,' all, and hence appears black. But till uerfectlv hkrfr ui - " 6. Theory of Exchanges (Prevost— 17ft2l _1>»tW t^ i*oo n j regarding emission of radiant enefgy Ze T^ „ ui -f ^w*? t a hlocfc of ice, we feel a senJrinn S ? ™ n i Wwn * e stLind being at about W lose mLI ~ v C .V M beCTOM om b0 <*3 * Ice, which is a a SS w5. ? iadaiVm . than il rece " phenomena. q * al dnd ma >' ** a PP lie ^ to all si,, V>teZ^ t ^ ( ?£p^ d ™ «*! ^en it ^ at the absolute becomes necessar tteate hnL it I - ■ ^perature, it temperature and the nSM^Ste Tth£ 2?" ^ A confine our attention to the *>n> t *t ln ™ S€Ctlon we shall stance and the tm'roundin^ V,\ *«&****, both of the sub- The experiments can be ^^p^SeVV S^ XI. LAWS OF COOLING bulbed mercury thermometer and enclosing the bulb in an evaa flask, the walls of which are blackened inside and are further ma - tained at a fixed temperature by means of a suitable bath. The rate of fall of temperature of the thermometer gives the rate of cooling. It was found that if we plot the rate of cooling - -^- as against die excess of temperature as abscissa, the curve is a : ight line, or in other words, -£ -H0-4) « ■ B n is the temperature of the enclosur he rate of cooling is proportional to the excess of temperature of the substance above the surroundings. This is Newton's law of cooling. The law holds- when the temperature difference is not large. Even the earlier experiments showed considerable deviations from the law when the temperature difference exceeded 40 a C. In order to find out a law which wiH hold for all differences ol temp;: of experiments were performed by Dulong and Petit who found that their results were given by a formula of the form _d0 ' tit = k{aO —a&o), (2) where > instant and k depends upon the nature oil the emitting e, and 0,6 O denote respectively the temperatures of the emitting . jd the surroundings. .Stefan* showed later that the results of Dulong and Petit be represented by the equation -£-• <*•-*». (3) where a is a universal constant, and T and T the absolute tempera- tares oi the body and its surroundings ; or in other words, the emissive t of a substance varies as the fourth power of the ah erature. This is Stefan's law and can be deduced frort] theo . 1 considerations (vide sec, 26). It is therefore the correct law of cooling for black bodies, Newton's law of cooling can be easily deduced from Stef; law when the temperature difference is small. Thus if the body at temperature T -f- 87* is placed in an enclosure at temperature T°K, the rate of loss of heat, per second by unit area of the body is a (T+8T)*-aT* - „T* / 1+"^ ) '- ^r 1 = ±vTHT neglecting higher powers of ST. It is thus proportional Co the tem- t.ure difference 87'. ♦Josef Stefan (1835—1893) was Professor oi Physics in Vienna. He dis- covered the law known by his tuutie, 288 KADIATIOK [CHAP, Generally for rough work in the laboratory ! law of cooling is employed Eor correcting for heat loss even when the body suspended in vacuum. As shown in Chap* VIII, § 30, if cooling takes place by natural convection in air, Newton's law not hold. In accurate work tct rate of cooling should be irved and not computed. Exercise*— A, body cools from 50°G to 40°C. in 5 minutes when trroundings are maintained at 20°C. What, will he its tempera- ture after a further 5 minutes? Assume Newton's law of cooling to hold. Nov,- for .: ^ 0, = 50 t=W t 9 = X .*, C - lag, 30 j £ = - ,-. log (0-20) =-kt + C. 5, (to be determined) . 1 . 20 5 ' ''" 30 •'■ log4 "itr = 2 logj M whcncc 6 *=M3*s] 8. Emissive Power of Different Substances — Preliminary Experiments. —Having n experimental study of how the emissive power with i let us find how the with the nature of the surface ig body, is pur- Leslie : ted with i i while I with the sul to be n both 1 to fall itio of led by Lite thermopile in Lh gives, In this missivitii mined, Lesl 9. Reflecting Power. — T.I tig power of difl mb- found by allowing the radiation from a Leslie cube to n Be in surface and the refle< , Next the thermopile is placed at the -surface. The ratio of ns in I Meeting powi rbis ^e wi th I radiation employed, and as earlier workers did not. employ monochromatic radiatit i lults are m i ach value. 10. Diathermancy. — Diachermanq with regard to heat is analogous to transparency in the case of light. Subsi which allow radiation to pass through them are called diatk those [o not, are "called atk\ us. The early experiments in this di performed by Melloni and I. Direct radiation from a Leslie cube fell on a thermopile produ deflec- tion ; next the experimental substance was introduced and the deflec- tion observed. The ratio of the two deflections gives the diathermancy XT. RELATIONS BETWEEN THE DIFFERENT RADIATION QUANTITIES 289 or transmission coefficient of the substance. Solids, liquids and gases were treated in this way. The best diathermanOi are rock- salt {Nad), sylvine (KC1) , quartz, fluorite mid certain other crystals, diathermanous, but water vapour and carbon dioxide show marked absorption. 11. Absorption. — For studying the absor] lids Melloni took a copper disc and coated one face with lamp-black and the other Tare with the experimental substance. The coppei dm was placed v,„ the L' abe and the thermopile, the Ian -black surface the thermopile. The plate will assume mperature T. ' al radiated per unit area of the plate H= (E + B)T. where E and E denote the emissivity of lampblack and of the substance. Now this must equal the radiation which it absorbs. The thermopile reading gi*es ET and therefore E. Knowing E and /,;■■ we g e i //. th< ii.sorbed. In this way the absorptive power of different si es could be compared. For gases a sensitive arrangement ed, terms diathermanous or athermanous lark in scientific precision. Every substance ought to be defined, so far as its trans- mitthij sorting properties are concerned, with respect to a parti- cular wavelength. In the above experiments monochromatic radia- were not employed. The whole subject is now studied as a branch of physical optics under the head 'dispersion and reflection' with which it Is intimately connected. 12. Relations existing between the different Radiation Quantities.— The foregoing experiments however show that at the same tempera- ture a lamp-black surface emits the maximum energy while a polished surface emits very little energy. It was further found that radiating and absorbing powers vary together; that good radiators are good absorl I pocr reflectors, while poor radiators are poor absorbers. Lamp-bin aque to radiation but allows radiation of very long wavelength to pass through it. A very simple experiment devised by Ritchie demonstrates vividly the relation between emissive and' absorptive powers of a body/ A Leslie cube AB, which is a hollow metallic vessel and ran be filled with liquid at any temperature, is placed between the two bulbs of a different; aeter (Fig. 8). The face A of the cube and the bulb D are coated with lamp-black while the face B and the bulb C can be coated with a layer of the substance whose emissivity is to be' investigated, my powdered cinnabar. By filling the cube with a hot liquid the index not found to move, a ace the amounts of energy received by C and D are equal. If e denotes the 19 C 6. — Ritchie's experi- ment RADIATION [CHAP. ►f heat emitted by the substance and a its absorptive power, lenote the corresponding quantiti. : amp-black, i e E a. w Now <?/£ may he called the coefficient of emission o£ the sub- atancej and hence the relation shows that the coefficient of emission ual to the absorptive power. It will be seen that the effect of temperature upon the coefficient of emission has been neglected here. Nevertheless the experiment shows at least qualitatively that the coefficient:? of emission and absorption vary in the same manner from one substance to another. These ideas were further developed and made more precise by Kirchhoff and Balfour Stewart. Before proceeding further it is how- ever necessary to define the concepts used in Radiation with more rigour. We now proceed to do this in the next section. 13. Fundamental Definitions. — If a body is heated it radiates Uons its surface in all directions, which comprises waves of all The nature of radiation depends on the physical properties of the boi denote by e\dk the amount of radiation measured js emitted normally per unit area per unit solid angle per second within the wavelengths A and \-\-d\. We shall call e ?i the emissive of Che body. arly if dQ\ be the amount of radiant energy a the body in the form of radiation (A to A + dk) and a absorbed by the body and converted is called the absorptive power of the body for these to A-f dX). For black bodies a\ =1 for all wavelengths, r other substances a. depends on the physical nature of the body. 14. Kirchhoff's Law.— In sec. IS « Chat emission and i vary together. In 1859, Kirchhoff* deduced an important which may be stated as follows : — %iven temperm ' ratio of the emissive power to the ''live power is the saute for all substances and is equal to the >er of a perfectly black body. Though this law was first recognised bv Balfour Stewart, the first to deduce it from therm odynaniical principles, apply it in all directions. It is therefore usually known as KirehhorTs law. We have considered here the total emission regardless of wave- Li but the same relation holds for each wavelength separately. radiations of the same wavelength and the same tempera- * Gustav Robert Kirchhoff (1824 — 1887), bora at Kfioigsberg; became Pro- fessor of Physics at E and Heidelberg, He h noted for his discoveries in spectrum analysis and the radiation law bearing his name. XI.] APPLICATION OF KIRCHIJOFF S LAW 291 turcj the ratio of the emissive and absorptive powers for all bodies is the same and equal to tl live power of a perfectly black body. This also holds for each plane-polarised component of any ray. Kirchhoff's own proof of the law is however very complicated and has. so I fallen into disuse. We shall give in sec. 19 a much simplified proof of the law. But before doing so let us consider its applications. 15. Applications of Kirchhoff's Law.— The law embodies two distinct relations, a qualitative and a quantitative one* Qualitatively, it implies that if a body is capable of emitting certain radiations it will absorb them when they fall on it. Quantitatively, it signifies that the ratio is the same for all bodies. Various experimental proofs and observations may be cited in support of die qualitative relation. If a piece of decorated china is heated in a furnace to about 1000°C and then taken out suddenlv in a dark room, the decorations appear much brighter than the white china, because these being better absorbers, emit also much greater light, If we take a polished metal ball and have a black spot on it by coating it with platinum black., then on heating die ball to about I000°C and suddenly taking it out in a dark room, it will be found that the black spot is shining much more brilliantly than the polished surface. Again take die case of a coloured glass. We know that green glass looks green because it absorbs red light strongly and reflects the green (red and green being complementary colours) . Hence when a piece of green glass is heated in a furnace and then taken out, it is found to glow with a red light. Similarly a piece of red glass is found to glow with green light. A more decisive example illustrating selective action is that of erbium oxide, didyrnium oxide, etc., which when heated emit certain bright bands in addition to die continuous spectrum. If now a solution of these oxides is made and continuous light, is passed through it, die very same bands appear in absorption. 1$, Application to Astrophysics. — B e s i d e s these applications Kirchhoff's law was in a sense responsible for the birth of two entirely new branches of science, viz., Astrophysics (physics of the sun and the stars) and Spectroscopy. We shall recount here briefly how these developments grew out of Kirchhoff's law, Newton had shown in 1680 that the sunlight can be decomposed by means of a prism into die seven colours of the rainbow, but Fraunhofer who repeated the experiment in 1901 with better instruments, found to his surprise, that the spectrum was not continuous, but crossed by dark lines. Their number is at present known to be 20000, but Fraunhofer noticed about 500 of them, and denoted the more prominent bands by the letters of the alphabet : — A, B, G, Such a solar absorption spectrum is shown in Fig, 9 (C) togeiher with a continuous prismatic (A) and continuous grating spectrum (B). T RADIATION CHAP, Frau nhofer never understood how these dark lines originated, neither did any oi Jus contemporaries. But he realized ilu i: great red them and catalogued their wavelengths. He examined the light from stai . and showed that then .p also crossed by dark lines just as tn the case of the sun. A tj stellar spectrum is shown in Fig. 9 (E) . In : antime, however, other sources of light, were examined by the spectres id some knowledge was obtained of their spectra and that a glowing solid gives a continuous spectrum, hut a flame tinged with NaCl pair of intense yellow Hues dark background. It was also found that, if an i discharge was I through a glass tube containing gas at low pressure, a large number of emission lines were obtained on a dark background, For hydrogen these are shown in Fig, 9 (D) . But still the dark lines of Fraunhofer remained unexplained. Some physicists, notably Ffeeau, o I that if the spectrum or the sun > - Id with the spectrum from a sodium flame, the yellow lines appear in the same place as the D-band of the FraunhoFej rum. Similar is the case with the hydrogen rum. in Fig. 9(C) ai I we see the emission spectrum of hydro by side with the Fraunhofer spectrum. It is clearly m some of the dark lines in the latter occur in the same place bright lines in the former. Th Iven by KirchhoS not only completely solved ; : r-reacning and extremely fruitful in its con- ned that the central body of the sun consists ich emits a continuous spectrum without I i lighl has to pass through a cooler atmosphere sur- mass. In this atmosphere all the elements like -c etc, :n ]i, in the gaseous form in addition to ' :. ::tc. W: : ri how the early workers studied the phenomenon of radiation with simple apparatus and arrived at some very general laws. Every substance when heated emits radiation, Le tJ light. Every substance has again got the power of absorbing light. Kirdihoif arrived at the same law from thermodynamic reasoning 19) and applied i(- to explain the dark lines. Sodium can the D-Iines when it is excited; hence when white light falls on it, it, can abs ; ' also the same light, and allows other light to pass through it unmolested. The gases in the outer cooler mantle round th therefore deprive the continuous spectrum From the central mass, of the lines they then ran emit, and give rise to the black lines. The D-bands therefore prove that there is sodium in the sun's at- mosphere. Similarly, the other dark lines testify to the presen their respective elements in the atmosphere of the sun. The correctness of Kirch hoi i matJon. is seen farther from flash spectrum results. We have supposed that 'the atmosphere of To face p. 292 CONTINUOUS PRISMATIC CONTINUOUS GRATING SOLAR ABSORPTION GRATING FED C B B a A (oTT rafw , , , | (D)l HYDROGEN BRIGHT LINE SPECTRUM OF ^-AURIGA (G r , TYPE) (E) ■ COMPARISON OF FLASH SPECTRUM (1905) WITH ROWLAND ATLAS (Flash photograph enlarged six times and Atlas reduced five limes). (F) m ii ii Aiiiiiiiliiiiiiiiiiiiiiii Fig. 9 Figs. (A), (B). {€), (D) have been reproduced from Know] ton, Physic flege Students, (E) from Muller-Pouilicts. Lehrbuch der Physik, Vol. V, part 2, and (F) from Handbuch der Astrophysik-, Vol. IV. I XI, APPLICATION OF THERMODYNAMICS TO RADIATION the sub contains Na. Now if we could observe the spectrum of the atmosphere apart from that of the central glowing core (the photo- sphere}, tlu: line uid appear I We cannot ordinarily do so because the solar atmosphere is so thin that we cannot cover up the disc properly, leaving the atmosphere bare for our observation ; further the scattered sky-light completely obscures the spectrum of the pari outside the dis during a total solar eclipse, the solar disc i ■ completely coveted for a short time by the disc of the moon, and skylight is also reduced by the moon's shadow. To anyone observing the sun through a spectroscope, the solar atmosphere will be laid bare al the time of totality and the dark lines will flash out as bright lines. This was Ei iii d actually to be the case by Young of Princeton in 1872, Fig. 9 (F) shows a Hash spectrum of 1905 eclipse which is placed side by side with the Rowland Atlas of Fraunhofer spectrum for the sake of comparison. The lines of the flash spectrum are thus found to be due to the elements in the sun's atmosphere. But Kii Hi ho |]"s disco voi}' is of much more far-reaehing import- ance than the mere success in explaining the. Fraunhofer bands would indicate. It clearly asserted for the first time, that ev& rent type of when it is properly excited, emits light of defcni wavelength which is characteristic of the atom. Just as a man is known by his voice, ot a musical instrument by the quality of its note, so each atom can be recognised by the particular lines it emits. Thus was bom the subject of Spectrum Analysis, which aims at identifying elements b) their characteristic lines, and forty new de- nts 'were added with its aid to the list already known. The different aton rded as so many different types of instruments, each capable of producing its own characteristic aetherial music. APPLICATION OF THERMODYNAMICS TO RADIATION 17. Temperature Radiation. — Since material substances at alj temperatures are found to emit radiation, it becomes possible to apply the laws of thermodynamics to the problems of temperature radiation. The expression "Temperature Radiation" should I clearly understood, for matter can be made to emit radiant energy in many way. othei than by heating, e.g., by passing an I '• trical dis- charge through it when in a gaseous state, by phosphorescence, fluores- cence, or by chemical action as in flames. But Kbchh oil's law holds only for temperature radiation. For radiation produced by other methods the law cannot be applied,! 18. Exchange of Energy between Radiation and Matter in a Hollow Enclosure. — Let us suppose that there is an enclosure with its walls, which are impervious to external radiation, maintained at a cons- tant temperature. We shall study what takes place when we place substances having widely different physical properties within this enclosure. In the first place the whole space is fdled with radiation which is being emitted by the walls. This radiation arises out of the heat 294 RADIATION * M??^ b ^ zi "^ «W««W*y squire the same temperature the walk* ihu can be proved by the method of nducfja ***. .tor suppose that inV equilibrium state the tem evSkLJi 11 a £a™°t engine w*T ^ used to a heat ESTSS T, ' W ? lls t0 A unU1 A J,ad die samc temperature as the walla. During the process a certain amount of heat will SmESJ 1 ^ - But ^ in A , would be reduced automatically i Si r /^J*™""- ;f according to our assumption, this is the .table state o/ affairs. Hence again the difference of at ure can he l hcdv J^ f* C0nVei ' SiOn ° f hCat f ° W ° rL We • **™ wk ' 7** ° 38SU ^ a temperature different from that of the g. we have at our disposal a means of continuous!, . ting the heat of a single body to work without mamtaininj conclude tfut ri materia] bodk: | J n the enclosure woi mately assume the temperature of the walls. fa vi^V'Vr' fPPosf that a body of hti [position £P '-Inn the enclosure. Then the different parts of iL'body" toe different emissive and absorptive . The total euro ab> " * equal to tile energ , S\iSfS temperature reman, llt [n the equilibrium , Z f t j£ T" »m PrTvost's 2J « Allows the * - « ****** is changed, And as the d'fS l" 1 ' Cerent coefficients of , ,,,. rhl ca"i ™ h mica! in qua, ..,., L cpiab of the enclosure or of any body plao [e it. T.ei L £ we have two enclosures A and B having walls of differs X w ^n?^ion u " fca ; ing f, h - cach *** tht " ?*s* ?h™ i?V xt of L WaveIe "St h lymg between A and A + o'X to pass through it. Now the walls of both A and B are mainlined at Z ™ jure T. Tf in the steady state the intenritv of radt at o^ ms thrown ?£ «f ater * A than in B. some fadiation will Thi Iv« and decrease in A. The screen is then closed The excess radiation will be absorbed by the walls of R and raise to temperature to r, while the temperature of thesis of TwTfaTL deduction of kirciuioff's law Now a Carnot engine may be worked between these two temperatures ding :l certain amount of work and lowering thereby ipe- ■ '■;' li and increasing that of A till the two are brought to the irature. The process can be repeated and thus we Duntol 'work while the entire system : cm colder. Thus in effect we are getting work indefinite!) by usii Meat of a single body. This is Impossible by the second lynamics. Hence the intensity of radiatioi the same in the two enclosures, i.e., the quality . of radiation depends only upon the temperature and on ii i thing else. If we now place a black body inside the enclosure, it will emit .in energy of the same quality and intensity as it absorbs. Hence it follows that the radiation inside the enclosure is identical in ev the radiation emitted by a black body at the same M perature as the walls ol the enclosure. These concha were rchhofE in I 19. Deduction of Kirchhoifs Law.— Suppose we 5 some sub ive and of wavelength lying between A and iid ax wly. We have alread a in the don that the amount of radiation dQ falling on the sub- die wall i depend upon the nature < hape of the walls. Of this a portion ot A dQ is I ■ nrbed b) remainder (1 -a A ) dQ, is reflected or transmitted. Further the su radiation equal to e^dX by virtue perature. Equating the energy absorbed to the energy emi ■ e x d\ = ax<i(l In the case of a perfectly black body of emissivity have (since a\ = I) , Combining (5) and (6) we get - F : — r. ; • (6) (7) ■;■ ratur the ratio of the emissive power to the dbsarp- ibstance is constant and equal to the emissive p of a perfectly black body. This is the thermodynamic proof oi Kirchhoftfs law. We have proved the law here for bodies inside the enclosure. Now since tire emissive and absorptive powei I only upon the physical nature of the body and ijui upon its sin i it follow-s that the law will hold for all bodies under all conditions Eot pure temperature radiation. II ■ 29G KA IU ATI ON [CHAP. 20. The Black Body,— The considerations put forward in sec. enable us to design a perfectly black body lor experimental purposes. seen that it an enclosure be maintained at a constant tern- are it becomes filled with radiation characteristic of a perfectly black body. If we now make a small hole in the wall and examine the radiation coming- out of it, this diffuse radiation will be identical with radiation from, a perfectly black emissive surface. The smaller the hole, the more completely black the emitted radiation is. Thus a correction has to be applied for the lack of blackness due to the finite size of the hole. This h due to the fact that some of the radiation coming from the wall is able to escape out and the state of thermodynamic equilibrium as postulated in section IS does not hold. This is almost completely avoided in the particular type of black body due to Fery (Fig. 11, p,' 297) . So we see that the uniformly heated enclosure behaves as a olack body as regards emission and if we make a small hole in it, the radiation 'coming out of it will be very nearly blackbody radiation. Again such an enclosure behaves as a perfectly black body* towards incident radiation also. For anv Tay passing 1 into the hole will be reflected internally within the enclosure and will be unable to escape outside, This may be further improved by blackening the Hence the enclosure is a perfect absorber and behaves as a ectly black body. Though Kirchhoff h cd in 1S5S that the radiation inside 3%ffi g ^a s f^,a3g-ESBSafefli j ^ Rheostat Fig. 10.— Black body of Wien. a uniformly heated enclosure is perfectly black it was long afterwards in 1895 that Wien and Lunmier utilised this conception to obtain a black body for experimental purposes, m The black body of Wien consists of a hollow cylindrical metal! t chamber C (Fig, 10) blackened-]- inside and made of brass or platinum, * Kirchhoff defines a black body as one "which has the property oi allowing all incident rays t enter without refk-etii u ;iii * them t:> leave again. See Planck, Wormcstrnhhing. | The walls however need not be black. Blackening merely enables the equilibrium state tu be reached quickly. XI,] RADIOME1 297 depending upon the temperature that it has to stand. The cylinder ated by an electric current passing through thin platinum foil as indicated bv thick dashes. The radiation then passes throng number of limiting diaphragms and emerges out of the hole O. The cylinder is. surrounded by concentric porcelain tubes- The tempera- ture of the black body is given by the thernio-element T. This is the type of black body now commonly used. Another type due to Fery is shown in Fig-. 11. Note the conical projection P opposite the hole O. This is to avoid direct radiation from the surface opposite the hole which would otherwise make the body not perfectly black, A striking property of such an enclosure is that if we place any substance inside it, the radiation emitted from it is also black and in- dependent of the nature of the body. Thus all bodies inside the enclos lose their distinctive properties. For a mirror placed inside will reflect the black radiation from the wall and hence the emitted radiation is black. Any substance if it absorbs any radia- tion transmitted from behind must emit the same radiation in conse- quence of Kirchhoff *s law and the total radiation leaving it must be- come identical with that from the walls i.e., of a black body. Fi.cr. 11- — Black body of Fery. RADIOMETERS 21. Sensitiveness of the Thermopile. — We shall now describe the instruments which are used for the measurement, of radiation. The differential air thermometer, which was employed by Leslie and the early workers, has already been described in sec. 2, but is now only of historical interest. Among the modern instruments, the thermopile has been des- cribed in sec, 2,. Special care however has to be taken in order to make it sensitive as we have sometimes to measure very small amounts of energy. In the design of thermopiles, the following 1 considerations have to be borne in mind : — (1) metals used should give large thermo- F..M.F. ; (2) junctions should be as thin as possible ; (3) connecting wires should be thin so as to minimise the loss of heat, by conduction ; (4) the junction should be coated with lampblack so that all the heat falling on it may be absorbed ; (5) it should be mounted in vacuum, so that there is no loss of heat by convection, and the deflections remain steady ; a rocksalt window is provided to let in the incident radiru The sensitiveness also depends upon the number of thcrmo- junctions but this cannot be increased indefinitely as the external 298 RADIATION CHAP. resistance will increase. The best procedure is to have the piles as- light as possible and to choose a number so that the total resistance of the thermopile is equal to the galvanometer resistance. The galvanometer to which the thermopile is connected should be o£ low resistance type with high voltage sensitiveness. For ordinary work the .suspended coil type galvanometers are generally employed but are not sufficiently sensitive on account of their high resistance. For accurate work the suspended magnet (astatic) type of galvano- generally employed, namely, the Broca, the Paschen and' the Thomson galvanometers. These are however very much suscep- tible to external magnetic disturbances and can be successfully used ' only by skilled workers. The Linear Thermopile — The extreme sensitiveness of the galva- nometers mentioned above enables us to work with only a few i of thermo-elements. The hot junctions are all arranged, in a vertical line. The wires are very fine and are wound" on n small frame. This is called the linear thermopile and is used for investigating the lines of the infra-red spectrum, 22. Crookes' Radiometer, — This consists of a number of thin vertical vanes of mien sus- pended at the ends of a light aluminium rod r inside an evacuated glass vessel (Fig, 12). Two such rods fixed at right angles are shown in the figure. They are suspended in such a can rotate about the vertical - a. The outer face of the vanes is coated b lampblack, while the inner faces are left clear. When radiation (thermal or light) falls on the blackened face the vanes begin to revolve in such a direction that the black- ed fare continually recedes away from the source of radiation. The cause of this motion is easily under- stood. The blackened face absorbs the Inci- dent radiation and thereby its temperature is raised, while the clear face remains at a lower temperature. The molecules bombard- ing the blackened face therefore become more hen ted than the molecules bombarding the other side of the vane and consequently they exert greater pressure on the vanes. The result is that there is a net effective force repelling the vanes from the incident radiation. This is known as the radio- meter effect, and is essentially due to the presence of molecules. It is obvious that the velocity of revolution will be a measure of the in- tensity of radiation. By suspending the vanes by means of a quartz fibre as in galvano- meters we can measure the intensity of radiation if we observe the . 12.— Crookes' radiometer. W xi.j THE BOLOMETER i .j Wi a radiometer has been employed steady deflection f^ft**;*^ ™X*#J of radiation. pUtinuSr strips or wires when .heaKd. ,,. thc ^native- P Tire WtiveaeB rf thebolo *«e r d^penc... ^ ^ fc ness of the galvanome er, W "° ™\ e the rise in .emperaM* and Figs. 13 and 14,-The Surface Bolometer. (2 ) linear bolometer to neueftt of distribution of energy in the spectrum of a black body. f evceedinsrlv thin strips of method of constructing such a thin conductor is as follows :— A sheet of platinum is welded to a thick sheet of silver and thc com- posite sheet is rolled. The sheet is then punched out as shown in Ficr 13, and attached to a hollow frame of slate. The ^lve dissolved off in nitric acid, tne end joints being protected by a coating o£ varnish. The strips are then coated with platinum black. Fitr 14, shows a front view ot such a grid. A grid so const ed has a resistance of about W ° hm For experimental purposes n^- fe^t e^'otte^'ntected in the form of a Wheatstone 800 RADIATION bridge. The method of connecting the grid is shown in Fig, 15. The grids 1 and 3 are in opposite arms of the Wheatstone bridge and are so arranged that the strips in 3 receive the radiation passing between trips in I and so die effect is doubled, 2 and 4 are similarly arranged but a] cted from radiation. The whole is enclosed in a box, In the absence of incident radiation the galvanometer shows no deflect ion. When radiation is incident on grids 1 and 3 defection i] oduced, In the linear bolometer a single narrow and thin strip of plati- num is as 24. The Radiomicroraeter. — ThJ vented by Boys and is essentially a thermocouple without an external nneter. A single loop c of fine copper or silver wire is suspended (Fig. 16'i between the poles of a strong permanent magnet NS as in a suspended coil galvan To the lower ends of the copper wire two thin bars of antimony So and bismuth Bi are attached, and the lower ends of these are again attached to a thin disc d or narrow strip of blackened copper. To the upper end of the copper lo is attached ;i thin h g carrying a light galvanometer mirror m few measuring deflection, the glass rod it. attached to a fine quartz pension q. The whole system is ex- uely light. Radiation, falling hori- i tally on the copper disc, beats the junction of antimony and bismuth and an current flows through the copper coil causing a deflection which depends upon the intensity of the energy re tivcd by the copper disc, To prevent disturbance due to diamagnet- ism of bismuth, Che rods of the thermo- couple are surrounded by a mass of soft' iron. The whole suspended sys- tem is enclosed in brass (shown shaded in the diagram), 25. Pressure of Radiation. — As ra- diation has been shown to be identical with light, it possess::-- :il the properties which are ascribed to light. One of the properties of light most important for our present purpose is it exerts a small but finite pressure on surfaces on which it is incident. This had been suspected by philosophers for a long time i he days of Kepler, who observed that as the comets approach •Reproduced from "The Theory of Heat" by Preston by the kind pcrmb > i .',',._■ <-:-■ j, I; .T-ill.i:-: & Cb. Fig. 16,* — Boys" Ri micrometer. PRESSURE OJ KAHM ' SOI the sun, the tail of the comet continuously veers round, so as to be >pposate the sun (Fig. 17). This he tried to explain on the assumption that light exerts pressure on all material bodies hich it is incident, but th< Lre increases in importance only the size of the particle is reduced. The die sun cometary matter, either dust particles or atoms, which are then repelled by light-pressure and thus lorm die tail. Though painstak- ing experimental investigations failed to show the existence ot pressure "it., Maxwell propounded in 1870 his Electromagnetic -Theory of Light, lowed that even on this theory Light should exert a ire but this is very small, being equal to the intensity divided by the velocity of light or Co the I jy-density. Calculation shows that the pressure due to sunlight is equal to 4-5 ;spercm* a Fig, 17,— Tail ot the comet' Bartoli also showed from thermodynamic considerations that radiation should exert some Finite pressure. The pressure of radiation is however so small that for a long time it battled all attempts to measure it. The difficulties were overcome only in 1900 when Lebedew, ant! a little later, Nichols and Hull demonstrated c mentally its existence and were able to measure it. They confirmed the theoretical conclusion that pressure is equal to the energy density of radiation. When the radiation is diffuse, it can be shown that pressure is equal to one-third the energy density of radiation. The fact that radiation exerts a finite pressure, however small it may be, is of great importance in the theory of black-body radiation. It shows tha ick radiation is just like a gas, for it exerts pressure %\[>. RADIATION CHAP. EXPLRIMENTAL VERIFICATION OK STEFAN S LAW SOS and possesses energy. In fact we can regard the black radiation as a thermodynamic system and calculate its energy and entropy, and apply the thermodynamic laws and formula:. We shall make use of these i in the next section. 26. Total Radiation from a Black Body,— The Stefan-Boltzmann L aWt — As already mentioned in sec. 7, J. Stefan, in 1 870., deduced kallv from the experimental data of Dulong and Petit that the total radiation from any heated body is proportional to the fourth power of its absolute temperature. In 1884 Boltzmann gave a theore- tical proof of the law based on thermodynamical considerations. lie red that the law applies strictly to emission from a black body. The law is therefore generally known as the Stefan-Boltzmann law, and may he formally' enunciated as follows : — If a black body at pemture T be surrounded by another black body at lute temperature T the amount of energy E lost per second per of the former is E = *{T*-Tf) t .... (8) where <r is called the Stefan's constant. For proving this law, we consider radiation in a black-body chamber and apply the thermodynamical laws to the radiation as mentioned in the previous section! Let u denote the energy density of radiation inside the enclosure, V its volume and p the pressure of ion. Then both U and p are simply functions of the absolute erature T. We h ther the total energy U of radiation equal to uV. Applying* equation (61) p. 267 we get &-t. ■ ■ ■ <9) and 1 = It tdU\ A u (10) or or du _ . tt the radiation is diffuse (sec 25) . Hence equation (9) reduces to T du u 3 ar "" T ,ir T * u = aT*, [11) where a is a constant, independent of the properties of the body. ce the total energy lost on one side by emission will also be pro- portional to the fourth power of the absolute temperature. That is the Stefan-Boltzmann law. 27. Experimental Verification of Stefan's Law.— The law was sub- jected to experimental test by various investigators, Lummer and ♦Boltzmann deduced the law by imagining the_ radiation to perform a Camat cycle. i :, -'d the conceptions of Bnrtoli. This is however unnecessary here for we have shown black radiation to be analogous to a gas and cm therefore :1 directly to apply the general thermodynatnieal laws to radiation, Pringsheirn investigated the emission from a black body over the range of temperatures lOCPC to 1260*0 and found the law to hold true within experimental errors, We give below a brief account of their apparatus and arrangement. A is a hollow vessel containing boiling water (Fig. 18) which acts as a standard source of radiation for calibrating the bolometer from time to time. The black body C employed for the ran; temperatures 2Q0°C to 600 °C consisted of a hollow copper sphere blackened inside with platinum-black, and placed in a' bath of a mixture of sodium and potassium nitrates which melts at 2I9°C. This salt bath could be maintained at any desired temperature. The temperature could be measured with a t henna-element T. 18. — Lummer and Pringshcim's- apparatus for verify trig Stefan's law. For temperatures between 900 'C and 1300 Q C the black body shown in Fig. 19 was employed. D is an iron cylinder coated inside with platinum-black and en- closed in a double- walled gas furnace. The temperature inside the iron cylinder was obtained by a therm o-e le- nient enclosed in a porcelain tube passing through the furnace. The measuring instru- ment shown at B was the surface bolometer of Lummer and Kurlbaum. A descrip- tion of this as well as the method of connecting it has been already given in sec. 23. Besides there are a number of water-cooled shutters so that the radiation can be stopped or allowed to fall at will. The procedure adopted was as follows: — The hath was heated up to the desired temperature and maintained steady, and then the Fir;. 19.— Bl: I-: , for 9£»* to 13D0 B C. 304 RADIATION CHAP shutter was raised to allow radiation from C to fall on the bolo- meter, and the maximum deflection registered by the galvanometer noted. The bolometer was kept at different distances from the black v and the inverse square law verified- Next observations were n with the black body at different temperatures. The observations were all reduced to a common arbitrary unit depending on the radia- tion from the black body A at 100°C. and kept at a dis ■ of 63$ . Tf d represents the deflection oF the galvanometer needle, T the absolute temperature of the black body, 290° the temperature of the shutter protecting the bolometer, then d =* *(r*-290*) 3 .... (12) from Stefan's law. The coefficient a was found to be constant. Hence the truth of Stefan's law is established. DISTRIBUTION OF ENERGY IN THE SPECTRUM OF A BLACK BODY 28. Laws of Distribution of Energy in Blackbody Spectrum.— From a study of the colour assumed by bodies when their D lire is gradually raised (see p. 281) it will be obvious that as the tem- perature of a body is raised, the colour emitted by it becomes rich in waves of shorter wavelength, Tn fact the wavelength for which the intensity of emission is maximum shifts towards the shorter wavelength 'side as the temperature is raised. These results were also arrived at. by Wien in 1893 from thermodynamic considerations n inside a hollow reflecting chamber. He showed thai contained in the spectral region included within the wave lengths A and A -+- dX emitted by a black at temperature T is of the form E^X=^f(XT)dX t ;i:-i and further if X m denotes the wavelength corresponding to die maximum emission of energy and E m the maximum energy emitted., vr-*, (i4) and E nl T~ & =B, (15) where B and b are constants. In other words, if the temperature oi radiation is altered, the wavelength of maximum emission is altered in an inverse ratio. Equation (13) is known as Wien's Displacement law. A further step forward in developing the theory was taken by Planck who showed from very novel considerations (which developed into the modern quantum theory) that the energy density of radiation inside the enclosure is given by V^isp^r* < I6 > where C t = 8-77/* c/A s and c^— ch/k = 4-96 b approximately. This is EXPERIMENTAL. STUDY OF BLACK-BODY l 30! K64 90 .known as Planch's lam. We shall 310-..- consider how the spectra can be obtained experimentally and the foregoing results verified, 29. Experimental Study of the Black-body Spectrum. — The first aatic study of the infra- lectrum was undertaken by La who illuminated the slit of a a is of sunlight and produced the spectrum by a prism of rocksalt. The rays were fen •. I lens on a bolometer which was arranged in n V me bridge adjusted for no deflection, he spectrum 1 be produced by a Ro grating but on account of the considerable Overlapp- ing of spectra o£ different m \ orclers prism 1 generally employed. Lew of quartz or fiuorite are m generally used. It is how- ever better to use a con- ^ - for focussing the radiation. Wien *s displacement was subjected to a scries of experim tests *••■- by Paschen, Lummer and fsheim, T< u I; c n : ; and Kurlbaum. We give below a 1 ption of the ex- periment and of E5-] Lummer and I EngsbJ im 1 ) . They used radii ictrically heated carbon tube and ced the spectrum by refraction * ; '- through a fluorspar prism. distribution of energy was measured by means I \ linear bolometer which was enclosed in an air-tight ci to diminish the absorption due to water vapour and carbon dioxide. Corrections were applied to convert the prismatic spectrum into a normal one by 1 means of the i. dispersion curve of fluorspar. The distribution of energy in the spectrum tor various temper: ■ sen 62! C 'K and lGiO^'K was obi and curves plotted (Fig, 20) . The ordinates • powers and the abscissa: w r avelenglhs. The full ■■ :.. Si,- 1 - ■> S Su, Fig. 20- — Distribution of fcjjcrgy in the 1 - kbody spectrum. 306 RADIATION CHAP. lines denote: the curves obtained experimentally while the dotted represent the curves calculated from a semi-empirical law | by Wiem The total radiation at a temperature is : •;• the area • een the curve and the -v-: varies as T 4 . The small patches of shaded area represent the aba ide and water vapour. ibsorption of fluorsp at G[t where the curves are seen to end abruptly* The wavel of maximum emission shifts towards the ' as the temperature rises. From these curves the values of E m and X m could be read. The experimental data are given in Table 1. Ta hie 1<—Experim gj ita I fe rification of Wien's Displaceme.! Temp. D K X m in f/s i-'ia A m T in ii lie ton e "ees 1017 1646 1.78 270.6 2928 2246 1460.4 2.04 145.0 2979 2184 1250.0 2.35 68.8 2959 2176 1094.5 34.0 2966 21 64 . 21.5 956 2166 908, 13.se 2208 723-0 4.08 . 2950 2166 621,2 4.53 2.026 2814 2190 i — 2940 2188 The i -ly the vi ions (14) and i the valu ws enable us to determine the of any sub'-:' -umed to be a black body) if X„ , be found out. The value of 6 is seen to be .294 cm X di \ | degree. This furni-' with a simple method of determining the tem- perature of the heavenly bodies. Thus Cor the moon X^ = 14^, and lience T= rT . \ i lx . ■=210°K. Similarly we can calculate the temperature of the sun (sec, PYROME1RY 39. Gas Pyrometers.- — In Chapter I we have already descri hydro :;i thermometer having a platinum- um bulb. For temperature above B00°C hydrogen cannot be used as the thermometric gas because the platinum bulb is permeable to it, hence nitrogen is invariably employed. As regards the material of the bulb Jena glass can be used to about 500*0 while quartz can stand up to about 1300°C but is attacked by traces at alkali from XL] RADIATION PYROMETER 307 hand etc. Ordinary porcelain is porous and permeable to the theimo- L but glazed porcelain was formerly employed up to about L000°C ; ' hich however the glazing softens and brea u its expansion is not regular. Platinum alloys are now invariably used. ln P Xv o< platinum and iridium was employed by Holborn and the bulb brittle. Holborn and Day found the alloy of 801 t-20Rh best £r the purpose u d a modified Lorn, ol this material to mt lfiOO'C. The} performed experiments with great rare and took It 25 - ;J L -, •; vor so that their value, are ed as standard at high temperatures. The correction however becomes enormous at these high temperatures and hence there is considerable uncertainty in determining the high temperature hxed points with gas thermometer. 31. Resistance Pyrameters.-The platinum resistance PF°"^er has been described in Chapter I. It can be W& to about U though the melting point or platinum is 1770°. It used above 1000 however, the platinum undergoes some change and does not return to its initial zero and has to be restandardised. _ The mica insulation ; , sometimes breaks due to moisture getting inside. 32. Thermo-electric Pyrometers.— The tbermo-ek ■ roraeters have been completely described in Chapter I. It is shown there that, for temperatures up to 600°C the base coup es O u-cons tanta n, i are the most sensitive. The Pt OOPt-lOUh couple i however best for all high temperature work and can be used to about A mo- lOCr couple up to 1500°C when the latter is much more sensitive, couple of Tr, QOIr-lORu can be used to about 2100*C while ther couples of tungsten and an. alloy of tungsten with molybdenum have been used up to 3000 °C. RADIATION PYROMETER 33. Temperature from Radiation Measurements.— In flu fore- went we can find its temperature. We way cither measure the total radiation emitted by the body and deduce the temperature by making use of Stefan's law (equation 8). These are called lotal Radiation Pyrometer*/ Or we may measure the energy i imiti :d in a particular portion of the spectrum and make use of Planck's law (equation 16) of distribution of energy in the spectrum. These are called Spectral or Optical pyrometers. Radiation pyrometers possess the great advantage that they can be employed to measure any temperature, however high that may be or wherever the object may be. The pyrometer itself has not to SOS RADIATION [CHAP. be raised to that temperature nor need it be placed in contact with the hot body. Farther there is no extrapolation difficulty as the radiation Eormube have been found to hold rigorously for ail tempera* lures. But the radiation pyrometer suffers from a serious drawback. It can measure accurately the temperature of ijodics only. It is, however, generally employed to measure the temperature of any hot source. In that case it \ at temperature at which a perfectly black body would have the same intensity of emission (total or spectral) as the body whose temperature is being measured. This temperature is called the . *erature of the substance and is conseqi lower in all cases than its actual temperature. The greater the departure from perfect blackness the greater is the error involved. The lower practical Limit for radiation pyrometers is about 600°C, for then the emission from substances becomes too small to be measured accurately. Still, however certain devices it can be used to measure much lower temperatures as that of the moon. 34. Total Radiation Pyroirtelcrs, — Tery was the first to devise a radiation pyrometer based on Stefan's law. These pyrometers are merely thermopiles so arranged that the readings are independent of rne dflsta itween the hot body and the pyrometer. We shall describe Very mirror pyrometer which is typical of this class. Fig. 21 shows a modern type of ToMilUfaaioH j mh Radiation incident from the right side falls on the concave or M which can be moved backwards and forwards for the purpose of focussing the radia- tion on the receiver S to which the hot junction of the thermocouple is attached. The cold lion of the thermo- couple is protected from radiation by the tov T and is further surrounded by the box B which also contains the thermocouple receiver, and a small opening just in front of S. The electromotive force developed is read on a mtilivoltmeter connected indicated in Fig. 21. The instrument possesses no lag. the steady state being reached in about a minute. To enable the observer to radiation pyron ipening being Left at the centre of the mirror to allow the incident radiation t. pnss. Now if the image of a straight line formed by the concave or does not lie in the plane of die inclined mirrors thev will form two images separated by a distance (see Fig, 22) and the line will si. OPTICAL PYROMETERS 309 appear broken when seen through the eyepiece. The concave mirror is moved till this relative displacement of the two halves of the linage disappears, and then the apparatus becomes adjusted. It will be en that so long as the heat image formed by die concave mirror is larger than the" hole, the thermocouple measures the intensity e heat image ot the total radiation. For if the distance of the object is doubled, the amount of radiation failing on the mirror r V:l-n Fi?, 22. — The focussing device. is reduced to one-fourth, but as the area of the image is also simul- Lisly reduced to one-fourth the intensity is unaltered. Thus the indications of the instrument are independent of the distance. Hence in actual use it is essential that the object, whose temperature is to be measured, should be sufficiently large and should be placed at not very great distance in order thai its image is always bie an the aperture in the box which in ordinary instruments is about 1,5 mm, in diameter. The E.M.F. of the couple in these cases is given by the relation V = a(T°-T l ) > .... (17) where T is the temperature of the black body to be measured, T Q the rature of the receiver S and b a constant which varies from 3,8 to 4,2 depending upon the instrument. Generally T can be negl in comparison with T. This departure from the index value of 4 is clue to various ran,:-;, It is for this reason that the pyrometer has to be calibrated by actual comparison with a standard thermometer j radiation from a blackbody chamber or heated strip. 3S. Optical Pyrometers, — In these, the intensity of radiation from a black body in a small width of the spectrum lying between X and A -|- d\ is compared with the intensity of emission of the sine colour from a standard lamp. The formulas required for this case can be easily worked out by assuming Planck's law. There are two typ , of optical pyrometers : — (1) the Disappearing Filament type, (2) the Polarising type. We now proceed to describe these instruments. 310 RADIATION [CHAP, XL] RADIATION FROM THE SUN 311 36. The Disappearing Filament Type.— This type of pyrometer trst introduced by Morse in America. It was later inrproi HTi and Kurlbaum, and by Mendenhall and . A pyro- r of this type is shown in Fig, 23 and is essentially a tele ig a lamp at the position usually occupied by the c a metal Lube containing tbe filament of a lamp L which is heated by the battery B and the current can be adjusted to any amount by rying the rheostat R. C Radiation from the source whose temperature is re- quired is focussed by the lens on the lamp T, where a heat image is formed. The lamp is viewed through the eyepiece E in front of which is pi need a red filter gl "ties there are a number of limiting rms. In periment the - looks through E and varies the current in the till the filament becomes invisible against the image of the source. If the current is too strong the filament stands out :;e current is too weak, th ' nt looks black. The filter g to be done for approximately m • ' | . ■ i i I [n - O ' : v,' nh -Li pie and then extrapolated by t rom the strength of the current required to matc l : ! the incident radiation can be directly calibrated in degrees. 37, Polarising Type.— In 1901 Wanner cons;. mother p) meter in which the comparison was made by the aid o£ a polarising Fig. tfiug Filament 'Pyro:-. L, Fig. 24, — The Wanner Pyrometer. device. Here the ray of a particular colour obtained from the source is compared with a ray of the same colour obtained from a standard electric lamp. A diagram illustrating the essential parts of the ins- trument is £iven in Fig, 24. a, b are two circular ho es arranged optical axis of the system. Radiation b the ; urce enters the ■■ trough a, while the comparison :n is supplied b-. which illuminates the rig I the latter directs the light on to b, Both beams are rendered parallel by means of an ach \c lens L 1( winch is ced ai a distance from J t equal to its focal length, lhe Llel beams nre dispersed by the direct vision spectroscope fc> and tirough the polarising Rochon prism R which separates each beam into two beams polarised in orthogonal planes. B is a biprism placed in contact with a second achromatic lei f n r, : | ams on the slit D 2 - The biprism produces in aos deviations of such amount that one image from each source is brought into juxtaposition. Since the holes a, b are at the focus of the lens L 3 the images produced by md lens are also circular but the biprism • into semicircles. Six out of the ei images are stopped out while the remaining two are observed through If the two beams are of equal intensity a uniforj nainated disc with a diametrical line is observed if the plane of polarisation of nicolpri : i 3 with the plane oE polarisai onent. Rota icol in either direction diminish^ : ;htness or one image and augments that of the other. If the ants are of unequal intensity n: [s affected by i i the nicol prism in either direction. A «a attac to the analyser ' ■ rve the angle Let the angle at which radiation from any source matches the lump be <#>. Then it: can be shown that In tan #-=fl-f--= . . • * ■ (IS) where 7" is the absolute temperature and a 3 b are constants. If we determine two values of $ corresponding to two values of T. we get 3 straight line from which the temperature for any value of $ can be d. In practice, the pyrometer is calibrated by a direct comparison ith a standard thermometer, and the disc is directly graduated in deg;" 1 RADIATION FROM THE SUN 38. The Solar Constant.— The sun emits radiant energy conti- nuously in space of which an insignificant part reach: rth. But even of this incoming radiation, a considerable portion is lost by .: and scattering by the terrestrial atmosphere and is sent back i the interstellar space/ The best reflecting constituents of atmosphere are water, snow and cloud. The scattering- is partly due to the dust particles and partly due to the air molecules and is gener- ,' small. Further the radiation is heavily absorbed by the earth's ■sphere, the total absorption varying from 20 to 40% depending RADIATION [CHAP. the day and the season of the year, and the different parts of die spectrum are absorbed to a different degree. We :. Eor a more constant quantity which is furnished by h solar radiations are received by one sq, cm, of surface held at right angles to the sun's rays arid placx the mean distance of the earth provided there were no absorptio the atmosphere or provided the atmosphere were not present. This is call . ant and is generally exp in calories minute i shall now give a method 'of determining the constant, 39, Determination of Solar Constant. — The Absolute Pyrheliometer. — Among the early workers who attempted > neaaure the constant were Wilson. The instrument; which the solar radiations arc measured are called pyrheliometers. We shall describe the mater-stir . . smpioyeti in the Astro- B Fig. 25. — The wa1 iter. physical Observatory of the Smithsonian Institution, Washington, a cross-sectional view of the apparatus. ilacfc-body chamber for the reception of solar radiation, whii further protected from air currents by a vestibule, not shown. This is simply a hollow cyli need in front of AA. The chamber is blackened inside and has its rear end of conical shape, ai surrounded by water contained in the calorimeter DD which is 51 nisly by means of a stirring arrangement BB run by an el motor from outside. C is a diaphragm of known aperture £< ing the solar radiation. The incident radiation is completely absorbed by the chamber producing a rise in temperature of the water contained in the calorimeter. This rise is measured by the platinum resistance thermometer F whose wire is carefully wound upon an insulating frame round AA; At E is inserted a' mercury thermometer. The XI.] DKIT-KMLXATION Ql' SOLAR CONSTANT SIS calorimeter DD is carefully insulated from thermal effects occurring ■ 1 : For calibrating the instrument, a known amount of electrical energy is supplied to the manganin resistance wire G and the rise in tempera t tire noted. Thu eat obtained by solar radiation can compared with the heat generated electrically* Another type of pyrheliometer is called the water- flow , meter. these a steady stream of water hows past the absorption chamber and die temperatur 'ween the incoming and water is observed. Instruments are called absolu they measure the enerj v. For most purposes it is more con- ient to employ secondary pyrheliometers in which radiation is ;rbed by a blackened silver disc. In the 'compensation pyrhelio- meter of Angstrom there are two thin strips, of metal idem:. I : every way which serve as the calorimetric body. One of these is sosed to the sun while through the other, which .is shielded by a screen, an electric current is passed. The strength of the current is regulated that the temperatures of the two strips, as indicated by thermocouples attached in opposition, is the same. The energy of incident radiation Is then equal to the electrical energy suppli Wing the length and breadth of the strips and their ion incident radiation per unit area can be calcula ith the help, of these instruments we are able to measure the rati: reived per minute at the earth's surface and from this w ? Iculate the solar constant The radiation received vari with the time oF the day, depending upon the zenith distance of the sun*| If we assume the absorption to be due to an atmosphere homogeneous composition, then applying- Biot's law, we have 7 = I, (TV, [l\ where d is the thickness of the medium traversed, I (> , I the intensity of (hi beam just at the beginning and the end of the medium. This d strictly for a monochromatic ray and a homogeneous medium. In t. e atmosphere there is no homogeneity in dust content or i densiy in a vertical direction. In the actual' e: periment the intensity of sdar radiation as received on the earth is oh liferent elections of the sun on the same day with constant sky conditions. rhei d varies as sec z where z h the sun's zenith distj 1 urther ling as a first approximation* that k t is the sara . ! wave hs we can write S - S,, a«« ■, (20) v., S represent the true and the observed solar constant'res- [y. and a is called the transmission coefficient and varies from 0.551 to 0.85. Then taking logarithm* assu lend Then taking logarithms log; S — IogS -psecz log a. I- or •:•.:• ate work h or a i, not assume 1 •o -.| w 8 **? 08 ™ se Parately and from this extrapolated curve the vali RADIATION [CHAP, Plotting fhe values of loo- S as ordinate* and the corresponding values of ser z as abscissa we obtain a straight line wjjose intercept on tne ate axis gives log 5,> whence £<i is found, . curate experiments give the mean value c \ the i snstant to he 1.937 i - per minute per sq, cm. The i int i found to .en-year cycle of the sun. 40. Temperature of the Sun.— The sun consists of a central hot linating in a surface called the sion 'temperature of the sun' we generally mean the temperature of -c The temperature inside the central core is, how milc h ! n this. We shall now describe some sin for de ig the temperature of the photosphere of the sun, on measurement of radiation. 41. Tamperatwe from Total Radiafiwu— According to cat serrations by Abbot and others the so mte per cm 2 . We can now find out what wouli ■ temjera- nrface free from an sre and havir s \ om ■ v.- which would emit the same tota ceof the earth, This naturally for the ture of the photosphere. Lei the radius of the sun be r, a a '- of heat loft by |" H — '\ir^<iT\. u. If we Lejcrjbe is R concentric with the sun (R being the d ill be spri ad over d per unit surface of the earth is 4irr 2 4'jtK 2 *T*, -=(f)v (22) Now r/R, the r an i ngulaT radius of the sun — 959" = 4.49 >( fans ; cr— 5,77 y 1 fr 5 ergs sec.- 1 cm." 2 degree-*: S = 1-957 cal. cm. -3 min."" 1 . Hence we obtain after substituting these vahss in that T — ain2 c }L. 42. Temperature from Wavelength of Maximum Emission.—- Wien's displacement law A M T = 0.2884 cm, X degree (eq i 14) also be employed to determine the temperature of the sun. Abbot's investigations show that A m z=475S A, U., wience T = G059--K. This temperature Is about 300° higher than flu tem- :• lire deduced from total radiation. Thii i gence is iasiiy understood if we remember the fact that contrary to il u otion made here, the sun docs not actually radiate like a blai I .iy. 43. Temperature of Stars. — The stars are so many suns, only they are at enormous distances compared to the sun. The detemina- XL\ TEMPERATURE OF STARS tion of their temperature is subject to the same uncertainties as in the case of the i Eollowed are. almost identical ex that her :otal radiation method fails as it is often no line the diameter of stars. The temperatures are :' intensity in their emission s] pplica- tion of equation (14), but the actual methods are far more complii The temp u ' : a|ltl othei ways - from 2501)'" C (red stars} to nearly 30,000° C (for bluish white st Books Recommended* i, GlazebrooTc, A Did' ,' Applied Physics, Vol. 4, Arti by Coblentz. 2. Planch, WSrmestmhlung or English translation by Mash ;). E. Griffiths, Methods of . • Temperature. ■f. Bur id Le Ch: telii , T " isurement of High 1 ure. Abbot, The Sun. CHAPTER XH THERMODYNAMICS? OF THE ATMOSPHERE 1. ] ii the present chap; tiall consider the application of the principles already developed in this book to the earth's atmosphere. e doing so, however, we shall first state .some of the results already known to us from the study of the atmosphere. DU ution of temperature 2. Vertical Distribution of Temperature. — The results of aerolo- observations show that the temperature of the atmosphere as we go up in the atmosphere. By combining several observations made over a given region, the form of the curve :i»2[C JI2208 Jos 20 Temperature*/* 1 UJvJuqo F ff-. c :cmpevelaT« taia =10*A 1 i- — Vertical distribution of monthly mean temperatures ever A g the variation of temperature with altitude is determined. This curve varies slightly with the season, particularly at the lower levels. the vertical distribution of the monthly mean temperature over Agi a (lat. 17° TO'N, long. 78° 5'E) in the Uttar Pradesh of l The fall of temperature due to a rise of 100 metres is usually called the vertical gradient of temperature and the fall of temperature Homed* rise in altitude is usually known as the lapse-mlz of temperature. The lapse-rate over any particular region varies with altitude and there is also a seasonal variation. 3. Troposphere and Stratosphere. — From Fig. 1 it is evident that Agra up to a height of 13 to 16 km. there is a rapid fall of rature, above which., however, the rate of fall dirnxni at a height of about 20 km. the temperature becomes almost constant. xii,] TROPOSPHERE AND STRATOSPHERE 317 i.€. t the lapse-rate vanishes. A similar discontinuity in the vertical distribution of temperature in the atmosphere is noticed all over the ■ :j.Id, although the height at which it occurs is not the same ei where. The outer shell of the atmosphere, in which the tempers remains practically constant with vari i of height, is. gi'mi ecial name ol •' foe Zone or Is to distinguish it from the Tr> re or the Convecth which is the lower portion in. which there is a considerable fall of tempera- ture with height. The surface o of these two region'- i the atmosphere, which plays a very important role in modern atmospheric circulation, is railed the tropopause. TV i he of the tropopause varies with the latitude. The Iropopau ■ is to lower towards the ground as we proceed from the equator to the poles, its heighJ being about M km. ii the equator and about 8 to 10 km. at the poles and is greater in summer than in winl : causes of the diminution of temperature with height in the troposphere are manifold. Let us try to explain here in general terms why this temperature diminution occurs. The solar radiation 1 its pas^ge through the earth's atmosphere Is only slightly al It, and the amount a isorbed d over such a larj air that the latter is not ■■■ heated by i radiation. In contrast to this, however, the energy received by earth is concentrated and therefore heats its surface considerably. ie heated surface in turn warms the air above it, parti) by contact and partly by the long wavelength radiated by it and absorbi •' '•■ the air. Now the temperature of afr at any height depends upon the 'total energy absorbed and emitted by it. The tower a ton | :i ^ sri temperatures emits more energy than it a and therefoi I toco I :■ radiation. These two phenomena, the h iti of the earth and the cooling of the layer above, so affect the density of the atmos- phere as to cause vertical convection, in consequence of which the warm ascending air becomes cooled through adiabatic expansion ami the descending air becomes heated by adiabatic com ires ion since the pressure or the atmosphere decreases with elevation. In tin's v the decrease of temperature with increase of elevation is established and maintained throughout the region in which vertical Jon takes place. We shall calculate in § 5 an expression for the lapse-rate. Above n certain height, however, convection becomes feeble ami the temperature of die atmosphere falls so touch that the heat radiated by it becomes equal to the amount absorbed by It either directly from the earth's radiation or from The p 'radiation. (This is be- cause the heat radiation received by the air from the earth reman practically constant at al] available altitudes) . The temperature of the layer remains practically unaltered at and above this b and therefore the convection currents cease above this heij I he stratosphere therefore is a result of the cessation of convection c rents and of establishment of radiative equilibrium. *This has already been discussed on pp. 311-313. 318 [IMODYNAWICS OF THE ATMOSPHERE ! 4, Vertical Distribution of Pressure.— The theory of the tton of pressure with increase of altitude is based on the applies of the gas laws Ei losphere. Lei: ua assume thai :re is in convei juilibrlum while the - e is in isothermal or ra equilibrium. Od account of convection currents the composition of *he air in the troposph ctically the same at ail I or the tl ical calcula; tion of pressure we can assume a mean gas constant for the atmos- pheric air- For the stratosphere hou \\ mvection currents cannot be assumed to exist since wa have assumed the layer to be isc hence no fixed value can be assumed for the gas constant R/M. We must therefore treat each constituent separately with its proper gas constant. Let -dp be the decrease of pressure corresponding to an increase vation dz. Then 'if p is the density of air at the point under consideration we have, on equating the decrease of pressure to che tit of die column dz f -dp^zgpdz. . ■ (1) From Boyle's and Charles' law we have h — pRT/M where M U die ■ [ght of the gas. Hence (1) yields (- ; ) RT m r ) p -' r\t * ■ ■ T to be constant, which is far from being the case as far posphere t on integrating i result P = pc r (•!) This is known as irmula and gives the pressure p at the height z in terms of the pressure /* ft at the earth's surra* As the temperature of the air column is really not constant, in actual practice the mean temperature, of the air column is substituted in equation (4) , This equation can then be readily applied to cal- culate the difference in height between two stations when the baro- metric pressures at the two stations are known. For calculating the distribution of pressure in the stratosphere we have to apply equation (4) to each constituent scp due to variation in composition M does not remain the same at all is. We therefore obtain ibr the partial pressures the result p = p t e-'W IT ;p=Po-e-* >i '< ! * T . ■ - (5) and the total pressure is the sum of these partial pressures, 5. Con vective or Adialwitic Equilibrium.— We have already in S 3 that the troposphere is mainly in convective equilibrium. Hence XI!.] WATER VAPOUR IN THE ATMOSPHERE 319 ■ ion (4) which was deduced on the assumption that the ;. phere I r . at rest ia not quite correct. The fact h really ■ : mines the distribution of the atmosphf isation oJ die condition tha on being moved from cue place to another, shall take up the sure in its new position without, any loss or gain of . taction. Tin: law • ting he volume and pre in the hould on such assumptioj :imate to the atic law. If the adiabatic law p = cp* holds for the atmosphere, we da dp A dp iz Hence from (1) we get cyp- -i* - Integrating- this v -: -<*-!)»,* (6) where p () is the de at zero level. This is the law according to which the density shoi id i 11 off with increase in height in the tropo- sphere. Since >RT/M, we get on substitution in the relation p = cfP the result T = R -or-1 R(Tr-T) ■ - ess titu ting this in (6) we obtain ■ T is the temperature at height zero, and J llie mechanical equi , alent of Heat. Now since y = c p je v wd JM {n p -r v ) — R, the : relation yields* ... (V) Jr 10293' Thus the temperature decreases proportionately to the increase of height as we yo upward;:; in th< re. Substituting numeric ■' ■ find that the constant of the above equation is about a. This value is about twice the experimentally observed tem- per;!; tamely 5°C. per km. WATER VAPOUR IN THE ATMOSPHERE 6. Hygrometric State of air. — Water gets into the atrai :i account of evaporation from surfaces of oceans, lakes, could be (25), p. -: : I :-i::;:lii n (23), S or THE vi -• CHAP. ered mountains, moist soil and, from various other sources. iration d pends upon a number of factor:-.;, na e rature of the air, eloci y, the pressure and the amount of i ' < i : litis. Increase of temperature ai wind velocity increase aporation while increase of pres- sure and of moisture in the atn decreases it. But the ;ity of the air to I, r vapour fa limited and depends upon temperature only, At a temperature i, air can hold only a o : pour which is giv< i the satun :ipour ure corresponding to the temperature t, and this anu it • of temperature. In Table i w ..i-ated vapour pressure of water at different temperatures and the values are Table 2. — Maximum vapour pressure of water in mill-'. of ' ent temperatures. D C o 1 2 3 4 5 6 7 8 y 4,579 5.2M 6.101 : . 7.0^ ?.SI3 8.045 10 9.21 11.23 • . . 19.8 23.76 , 37.73 39.90 42. IS ■•v.-'/ 49.69 2 •1 12 79.60 #8.02 6 8 10 14 16 30 1 3.8 149.4 163,8 179.3 196,1 214.2 A 327..1 355.1 S 450.9 ■:\ . 610.9 6^7.6 7C7.3 760.0 875.1 937.9 1004,4 plotted in Fig. 2, p. 321. If the air contains the maximum amount of water vapour that it can hold, it is saturated ; if it contains a k amount, it is unsaturated. In some cases, it may contain more than the equilibrium quantity; it is then called supersaturated. The eld in air may be expressed in gra a : c cubic metre or in tern essure in millimetres which it exerts. This is known as the "Absoluts Hum ic air. Humidity thus con, information regarding quantity of water vapour in the atmos The state of the ihere with regard to its actus ttent is generally expressed by its Relative Humidity, which is tin ctual quantity i pour present in a divert quantity of air to the maximum quantity that it could hold if it were sal 'he observed temperature. Relative K in • the hygrometric state of air Le., its moisture itent and hold moisture ran be fully specified by ire and relative humidity. 7. Dew-poir.. — If air containing moisture is proj cooled, a temperature wilt be reached at which the moisture that it contains X1L] HYGROMETER 321 IQ 20 SG 40 50 60 10 W Wlw-C Fig-- 2- — Vapour pressure euirve o£ water. is sufficient to saturate it. This temperature is called the Dew-point. Any further cooling of the air will bring about a deposition oE moisture on the surface of the containing vessel in the form of "dew". In the large-scale phenomena occurring in the atmosphere the deposition may take any one of the different 'forms, viz., Bog, cloud, rain, frost., hail, snow, dew, etc:. It is easily seen that the four quantities — tempera- ture, absolute humidity, relative humidity and dew-point are inter-related and a determina- tiorj of only two of them is suffi- cient. This can be easily done with the help of Table 1. Thus if the temperature and the dew point are 30" and 20°C. res tively, the saturated vapour iure at 20 Q C, is 17.54 nun. which gives the absolute humi- dity of the air, and the re] tive humidity (tt.H.) is (17,54/31.82) X 100 — 55%. Conversely., if the absolute 'humidity is known to be 9.7 mm. the dew-point is seen from the table to be equal to 10°C, and the R.H.= (9.7/3U2) ;< 100 = 30%. 8, Hygrometers. — The study and measurement of moisture pre- sent in the atmosphere is called Hygromeiry and the instruments used for measuring the amount of moisture are called hygrometers [Hys?o — moisture, meter = measurer) . From what has been said above, it will be evident that besides temperature we need measure any one of the three quantities — absolute humidity, relative humidity and dew-point. This gives rise to a variety of hygrometers which may- be broadly classified as follows:— (1) Absorption hygrometers such as the chemical hygrometer, (2) Dew-point hygrometers, (3) Empiri- cal hygrometers such as the wet-and-dry bulb hygrometer and the hair hygrometer for which no complete theory has yet been worked out but which are by far the most important. 9. The Cbemical Hygrometer, — Absolute humidity is directly found out by means of the chemical hygrometer. In this instrument a stream of air is aspirated slowly over drying tubes, and the gain in weight of the tubes and the volume of air passed over are recorded. Thus the mass of water vapour actually present in a given volume of air is found out, and this is compared with the mass required 10 saturate the same volume of air at the same temperature. Tin's biter quantity is given in tables. The disadvantages of this medio apparatu 21 S22 THERMODYNAMICS OF THE ATMOSPHERE CBAP. and not easily portable, the experiment takes considerable time and laborious corrections must be applied. Its chief use is for standardis- ing the simpler types of instruments in the laboratory. ID. Dew-point Hygrometers.— Hygrometers in which humidity is found from a direct determination of the dew-point are called Dew- point Hydrometers. Examples of this type are the Daniel!, the Regnault'and the Dines* hygrometers. The essential principle under- lying all ol them is the same, viz., a surf ace exposed to air is steadily cooled till moisture in the form of dew begins to deposit on it. The temperature is again allowed to rise till the dew disappears. The mean of the two temperatures at which the dew appears and dis- appears gives the dew-point. These hygrometers are however rarely used in meteorological work. They differ from one another in the manner of cooling or in the nature of the exposed surface. We shall therefore describe only one of them, viz., the Regnault's hygrometer. 11. Regnault's Dew-point Hygrometer.— This consists of a glass tube fitted with a thin polished silver thimble S (Fig, 3) containing ether. The mouth of the tube is closed by a cork through which passes a long tube going to the bottom of the ether, a thermometer with its bulb dipping in the ether, and a short tube T connected on the outside to an aspirator. When the aspira- tor is in action air is continuously drawn through the ether producing a cooling and the temperature of the thermometer falls. The pto- cess is continued till moisture deposits on the surface of the thimble, and the corresponding temperature is noted. In order to help in recognizing the first ap- ance of this moisture by com- parison, a second similar tube pro- wl th a .silver thimble S but without ether is placed beside it. Next the aspirator is stopped, the apparatus allowed to heat up and the temperature when the dew disappears is noted. The mean of these two temperatures gives the dew-point. The dew-point hygro- meter has the following disadvan- tages ;— (1) It is difficult to deter- mine the instant when dew appears, (2) the temperature of the ther- mometer does not accurately represent the temperature of the surface, (B) the instrument should be used in still air, (4) the observer him- self, being a source of water vapour, is likely to disturb the readings. Attempts have been made to minimise or eliminate these difficulties, but it is not possible to get continuous records from such instruments. Fiff. 3.— Regnault's Dew-point Hygrometer. XII.] WET AND DRY BULB HYGROMETER S23 12. The Wet and Dry Bulb Hygrometer or P*yehromet«r.— "Relative Humidity" can be easily measured by means of a wet and dry bulb hygrometer. This consists of two accurate mercury thermo- meters suitably mounted on a frame. Round the bulb of one of these is tied a piece of muslin to which is attached a w T ick extending down into a vessel containing pure water. The evaporation from the large surface exposed by the muslin produces a cooling and thus the wet bulb thermometer records a lower temperature than the dry bulb thermometer. In the steady state there is a thermal balance between the wet bulb and the surroundings. The greater the evaporation the greater will be the difference in temperature between the two. Now evaporation will be greater the lesser the humidity of the air and thus the difference in temperature between the wet bulb and the dry bulb is a direct measure of the humidity. The rate of evaporation is, how- ever, further affected by the pressure and the wind; large pressure tends to retard evaporation while large wind velocity accelerates it The effect of pressure is however very small and may be neglected, while the effect of wind is rendered constant by maintaining a constant supply of fresh air. A relation between the readings of the two thermometers and certain other quantities can be easily found. If T, T' denote the absolute temperatures of the dry and wet bulbs respectively, p the pressure of water vapour prevailing in the air and p* the saturated vapour pressure at T' s and 11 the barometric pressure, the rate of eva- poration will be proportional to £-p H and also to (T - T) ; therefore p'-p = AH(T-T'), (8) where A is some constant depending upon the conditions of ventilation and is determined from, a large number of experiments. In actual practice Ilygrometric Tables have been prepared by the Meteorological Office assuming the value of A for a fixed draught of air. From these tables the pressure p of water vapour prevailing in the air can be directly read if the dry bulb temperature and the difference between tlie dry and wet bulb temperatures are known. Knowing p the rela- tive humidity can be found, 13, The Hair Hygrometer. — For ordinary purposes the relative humidity can be roughly measured by the Hair Hygrometer. This consists essentially of a long human hair from which all oily substance has been extracted by soaking it in alcohol or a weak alkali solution (NaOH or KOII) . When so treated die hair acquires the property of absorbing moisture from the air on being exposed to it and thereby changing in length. Experiments have shown that this change in length is approximately proportional to the change, between certain limits, in the relative humidity of the atmosphere. Fig, 4 shows a THERMODYNAMICS OF THE ATMOSPHERE CHAP. hair hygrometer. The hair h has its one end rigidly fixed at A while the other end passes over a cylinder and is kept taut by a weight or spring. The cylinder carries a pointer whidi moves over a scale of relative humidity graduated from to 100, The changes in length or the hair due to changes in humidity tend to rotate the cylinder and thereby causi a motion of the pointer. The instru- ment must be frequejjtiy standardiz- ed by comparison with an accurate hygrometer and then its readings are reliable to within 5%. Kg. 4. — Tl meter. 14. Methods of Causing Con- densation. — We shall now find out under whaf conditions the water vapour present in the atmosphere be precipitated from it. This water vapour can be condensed into liquid water or solid ice if the actual vapour exceeds the maximum vapour pressure corresponding to the exist- ing temperature, This happens al- i exclusively when the air is cooled down more or less suddenly but in rare cases it may occur if the jure happens to increase ■lIill: to some local h as compression of saturated wate etc. The cooling of air may take place by the following three processes : — (1) Due to radiation of heat or due to contact with cold bodies. (2) Due to the mixing of cold and warm air masses, (3) Due to adiabatic expansion caused by sudden decrease of pressure. The first process should have been the most effective in produc- ing precipitation had it been active in large masses of air. But air even when it is moist, is a poor conductor and radiator of heat, so that radiation and conduction of heat play a minor roTe in the pheno- menon of precipitation. The result o£ the loss of heat by radiation or by contact, with cold bodies, such as the surface of the earth in winter, cold walls, stones, etc, is the formation of mist, fog, dew, etc. The second mode of condensation depends essentially on the experimental fact that the saturated vapour pressure of water increases much more rapidly with increase of temperature than the temperature itself. Thus if two equal masses of air, initially saturated at tern- XII.] ADIABATIC CHANCE OF HU.MlI> AIR 325 peratures t and ¥ respectively, are allowed to mix together, they will acquire the mean temperature t m = (t -|- F)/2, while the mean vapour pressure will be (e + e*) /2, where e, e' denote the saturated vapour pressure of water at temperatures t and t' respectively. On account of the above property, however, this mean vapour pressure will be greater than E, the saturated vapour pressure at t m and therefore the excess of water will condense. As an example take the following illustration : — Let us have equal masses of saturated air at 4° and 82°C. When mixed up the temperature becomes 18°C. To saturate the mass we require 15.4 gm. per cubic metre. The separate masses contain 6.4 and 33,8 grams and the mixture ,/' - gm. per cubic metre. Hence 20.1 — 15,4 = 4.7 gm. will separate by condensation. If the two masses of air are not saturated before mixing, there may be condensation in some cases. This will depend upon the proportions of the mixture. If both the masses are very near the point of saturation, then condensation may take place at some places, and no condensation or even evaporation at others. This explains the formation and disappearance of certain kind of clouds. This third process is the most important because it is active on a large scale and produces cloud and rain. When moist air is allowed to expand adiabatically its temperature falls and some of its moisture is condensed if the temperature falls below the dew- point. This is the process which generally takes place in the atmos- phere. An ascending current of moist air suffers a decrease of pressure as it ascends ; it therefore expands almost adiabatically and partis with some of its moisture. To calculate the cooling we have mass of air, IS, Adiabatic Change of Humid Air,— From the first law of thermodynamics if dQ be the amount of heat supplied to a given mass of air, «-*JT--}*-**T- (9) In case of a mass of saturated air rising upwards, the heat dQ is added as a result of an amount dm of vapour being condensed. Hence, d(l= -Ldm, (10) where L is the latent heat of vaporization. Therefore —Ldm = CpdT — '' ' MpJ (II) The total mass m of water vapour in the air per c.c« is given by m = 0*623— xp, P THERMODYNAMICS OF TILE ATMOSPHERE [chap- where 0.623 is the ratio of the molecular weight of water vapour to the weighted mean of the molecular weight of the constituents of dry air, e the vapour pressure, and p the pressure of the dry air and p its density. Hence dm de dp m e p Substituting this value of dm in (11), we have de- I.m . RT -Lm T + - dp - H dT+ j^jdp - 0, (12) or, Now dp=i -pgdz = —jppdz* Substituting this value in (13) we get dT = _ s \ RT ^ Jf dz , Lm de e >+-TdT (14) This is the rate of decrease of temperature with elevation of saturated air. All the quantities on the right-hand side of this equation are known, so that — — can be easily evaluated. Books Recommended. 1. Humphreys, Physics of Air. 2. Brunt, Meteorology. 3. Lempfert, Meteorology, 4i Hann, Lehrhuch der Meteorologie. 5. Wegener, Thermodynamik der Atmosphere. APPENDIX 1 ERRORS OF MERCURY THERMOMETER AND THEIR CORRECTION As mentioned on p. 2 various corrections must be applied to the mercury thermometer if it is used for accurate work. The method of applying these corrections is explained below : — (i) Secular Rise of Zero. Glass is to some extent plastic and therefore its recovery to its original volume is an extremely slow pro- During the construction of the thermometer the glass is heated to high temperatures and then allowed to cool. In this cooling pro- cess the contraction of the glass first takes place rapidly and then slowly even upto several years. Naturally therefore when calibration of the thermometer is usually undertaken the glass has not contracted to its final steady volume and the zero-point shows a secular rise for years due to this gradual contraction. This defect can be greatly removed by choosing suitable material for the glass of the thermo- meter, by properly annealing the tubes and storing them for years before making thermometers out of them. (it) Depression of Zero. This defect is also due to the defect in the property of glass mentioned above. When a thermometer is suddenly cooled from 100° to 0°C, the bulb does not at once regain its original volume and there is a consequent depression of zero, - magnitude is greater the higher the temperature to which the ther- mometer was exposed and the longer the duration of this exposure. The method adopted by the Bureau International to correct for the depression of zero is the "movable-zero method" of reading temper a- tures. In this method the boiling point (100°C) is first determined and immediately after, the ice reading is taken ; let these readings on the thermometer be X and Z respectively. Suppose this thermometer reads X s when immersed in a bath at (°C. Immediately after this the thermometer is immersed in ice ; let its corresponding reading be Z ( , Then the correct temperature f°C of the bath is given by x-Z 100, (Hi ) Errors in the fixed points. For the lower fixed point the thermometer is clamped vertically with the bulb and a little part of the stem surrounded by pure ice mixed with a little quantity of distilled water. Suppose in the steady state the mercury stand's at -0.PG ; then the freezing point correction Is -j-0J o C (additive). If the level of mercury stands above the zero degree mark, the correc- tion is subtractive. For the upper fixed point the thermometer is kept suspended inside a hypsometer with the bulb exposed to steam in the inner chamber. The steady reading of the mercury level is observed and the reading of the manometer indicating the pressure of steam noted. Suppose the thermometer reads no .2°C"under a pressure of 75.8 cm 328 APPENDIX of mercury. Since the boiling point for this pressure is 99.93°C, the correction is 99.93 -99.2 = 0.73°G and is positive. If the obse boiling point i.s above the calculated one, the correction is negative. (jV) Correction fornon-itnift ire. As capillary tubes are drawn and not bored, slight inequalities in the diameter of the bore are bound to exist in the atem of the thermometer. This necessi atea a small correction which is carried out as follows : — A small portion of the mercury thread is detached from the rest and its Le measured when it occupies successively different parts of the stem, say between and 10, 10 and 20, ♦ . , 90 and 100 marks. The measured lengths will vary from place to place due to non-uniformity of the bore ; let these lengths be t it l 2 lw> respectively. Let the corrections to be made for non-uniformity in the vicinity of the 0, 10, 100 mark be a^, «ki «iocj respectively. If I is the accurate length of the mercury thread used, J = / x -t- a 10 - % . . . . (1) J = 'a + *ao-»io ■ ■ • • ( 2 ) I = JiO + «1W» - «B0 ■ • • (3) Adding up, 10 I = & -f- J s + r 10 ) H- C^joo - flo) • • • (4) a 1OT and a n are the corrections to the upper and lower fixed points which can Trained experimental!) as explained in (Hi) above. Hence / can be calculated from ('!) . Substituting this value of / in (1) «,o can be calculated since a (t is known. Similarly from (2) a then di awn with the marked divisions i . a vil ..... a vl[ as ordinate. From this -.i Lures can be easily rend, In li thermometers this tedious correction has not to be applied by the user as the interval between the fixed points is subdivided not into equal parts but into equal volumes to represent the degrees on tins thermometer. (v) Correction for lag of the thermometer. If the bulb of a thermometer is placed in a hot bath, the thermometer will not attain the temperature of the bath instantaneously will require a small, definite interval of time to attain that temperature. This is called the "lag" i I the thermometer. The lag of the mercury thermometer increases with the mass of mercury and the thickness of the glass and also depends upon the nature of the medium surrounding the bulb. Due to this lag the thermometer reading will be higher when a bath is cooling and lower when the bath temperature is rising. Suppose we consider the case when the temperature of the bath is rising. The correction for lag is applied as follows: — The bulb of the thermometer is immersed in the bath and the thermometer read- ings noted as a function of time and the observations plotted on a graph with temperature as ordinate and time as abscissa (Fig. 1) . The curve AB represents the rise of temperature with time. To ERRORS OF MKKCLRY THERMOMETER : 'J find the correct temperature of the bath at each point on the curve, an auxiliary experiment is performed in which the bulb of the thermo- meter is immersed in a thermostatic bath maintained at a temperature somewhat higher than the maximum recorded in the main experiment. The thermometer readings are read at short intervals until the thermo- Tlrne A" ^ Time Fig. 1 Fig. 2 meter attains the steady temperature of the bath. These readings are plotted on a graph with time as abscissa and the difference of the h; Miometer reading and the temperature of the bath as ordinate, and the curve RS (Fig, 2) obtained. I l will now be assumed that the lag does not depend upon the actual temperature of the bath but only upon the rate at which the temperature is changing. Consider a point M on the main curve. The lag at this instant depends upon the slope (dO/dt) of the curve at this point M. Now find out a point P on the curve RS where the slope is the nine as the slope at the point M, Then the lag PQ will also be the lag at M. Hence if PQ s= MN, N gives the correct temperature corresponding to M. In this way the corrected curve A'B* can be easily drawn. (vi) Error due to changes in the size of the bulb caused by variable internal and external pressure. For diminishing the time lag the bulb of the thermometer is usually made thin, Am increase in external pressure therefore easily alters the volume of the bulb and causes a rise of mercury level in the stem. Suppose the thermo- raduated when the external pressure is equal to the atmos- pheric pressure, If the external pressure is now* increased the bulb will contract and the mercury En the stem will rise. The external pressure coefficient is defined as the ratio of the rise of mercury in the expressed in degrees, to the increase in external pressure, expressed in mm. of mercury. This ratio can be easily determined experimentally. Knowing this and the external pressure to which the bulb is subjected at the time of reading the thermometer, the correc- tion to be applied can be readily calculated. When a thermometer has been graduated in the horizontal posi- tion and a subsequent reading is taken with the thermometer in the vertical position, there is an increase in the internal pressure due to the vertical column of mercury in the thermometer. The bulb con- 330 APPENDIX sequently expands causing a depression of mercury in the stem, for which a correction is necessary. To determine this correction the readings of the thermometer are observed in the horizontal and vertical positions at any one temperature. The difference in the two readings gives the depression of mercury in the stem due to an increase in pressure caused by the mercury column which ex Lends from the centre of the bulb to the mark on the thermometer at which mercury stood in the vertical position of the thermometer. Expressing this pressure in mm. of mercury we can define the internal pressure coefficient as the ratio of the depression of mercury in the stem, expressed in •rces, to this increase of internal pressure. Knowing this coeffi- cient, which is constant, the correction can be calculated for any read- ing of the thermometer in the vertical position. (vii) Error due to capillarity. The surface tension of mercury causes an excess of pressure within trie meniscus over that outside. This excess pressure depends on the radius of the tube at the point where the meniscus lies {p cc 1/r). If the stem is not uniform in bore, there will be variations of internal pressure as the thread of mercury rises or falls and therefore the thermometer readings will not be very accurate. Further the angle of contact beween mercury and the sides of the tube depends upon whether the mercury is rising or falling, the meniscus being flatter when mercury is falling. Therefore a rising thread always gives somewhat lower readings than a falling one. it is also found that the mercury thread is less disturbed by capillarity when it rises than when it falls and therefore it. is preferable to take readings with a rising column. (viii) Error due to exposed stem or emergent column. Generally when the temperature, of a bath is measured, only the bulb of the thermometer and a portion of the stem are immersed in the bath. Tn such rnses the part of the stem exposed to the atmosphere does not acquire the temperature of the bath and therefore the thermometer reading will be less than the true temperature of the bath. The correction for this exposed or emergent column is applied as follows: — Let the thermometer reading he \ when the stem upto t 2 mark is immersed in the bulb. Thus n (■= f a — 1 2 ) divisions are exposed to the atmosphere, and its average temperature to is measured by a special integrating thermometer with a long bulb placed near it, with its centre coinciding with the centre of the exposed part. Let t denote the corrected temperature and B = m — g the coefficient of apparent expansion of mercury in glass (m = expansion coefficient of mercury, g, of glass) . Then a mercury column w T hose length is n divisions at * has to be corrected to the temperature t. If this column were to rise in temperature from t to *, the increase of height (which measures the increase of volume) would be «*{*—/<>) divisions. This must therefore be added to the observed reading t t to give t, Hence QT f a — n8t 1—nB EXAMPLES* I 1. Discuss the advantages of using one of the permanent g as a thermometric substance for defining a scale of temperature. Describe some convenient and accurate form of gas thermometer, explain its mode of use and show how the temperature is calculated from the observations made with it. (Madras, B.Sc.) 2. The pressure of the air in a constant volume gas thermometer is 80.0 cm. and 109.4 cm. at 0° and 1()0°C, respectively. When the bulb is placed in some hot water the pressure is 94.7 cm. Calculate the temperature of the hot water, 5, An air bubble rises from the bottom of a pond, where the temperature is 7°C,, to the surface 27 metres above, at which the temperature is 17°C Find the relative diameters of the bubble in the two positions, assuming that the pressure at the pond surface is equal to that of a column of mercury of density 13.6 gm. per cc. and 76 cm. in height, 4. Explain clearly the meaning of absolute temperature of the air thermometer scale. A gram of air is heated from 25 °C to 70°C. under a constant pressure of 75 cm. of mercury. Calculate the external work done in the expansion given that the density of air at N.T.P. is 0-001293. 5. What is an air thermometer? Explain the method ol measuring temperature by Callendar's compensated air thermometer, Describe a method for measuring very high temperatures. (A. IT., B.Sc.) 0. Describe the Callendar's compensated thermometer and explain how temperatures are taken with it. (A. LL. B.Sc, 1944 : Utkal Univ., 1952; Punjab Univ., 1954.) 7. Describe briefly the method of standardisation and the range of usefulness of platinum resistance thermometers, and discuss some of the difficulties of precise resistance measurement and the precau- tions to be taken to avoid or correct for these, (A. U„ B.Sc. lions,, 1929.) 8. Give an account of the construction and use of the platinum resistance thermometer, pointing out any special advantages of the instrument. (Utkal Univ., 1950; Gujerat Univ.. 1951 ; Punjab Univ., 1956.) 9. Describe various methods of measuring high temperatures. (A. U., B.Sc, 1931, 1932, 1949; Gujerat Univ., '1951; Punjab Univ., 1956.) 10. State, with reasons, the type of temperature measuring device which you consider most suitable for use at temperatures of (a) — 20()°C. : (London, B.Sc.) (b) — 50*G„ (c) 50°C., (d) 700°C> (e) 200 a C. ♦These examples have been classified and arranged chapterwJse correspond- ing to the twelve chapters of the book. 332 EXAMPLES 11. Describe two methods for measuring high temperatures. State: clearly the principles underlying them and the range and sensiti- vity of each. (A, V» B. St, 1941.) 12. Write a short essay on the measurement of (a) high and (6) low temperatures. (Delhi Univ., 1954; Punjab Univ., 1946.) iS. In what respects is a constant volume gas thermometer superior to the constant pressure gas thermometer and the mercury- in-glass thermometer. Describe the construction and use ot the inter- national standard hydrogen constant volume thermometer. (Patna Univ., 1949; Utkal Univ., 1954,) II 1. Enunciate Newton's law of cooling and show how corrections can be made for the heat lost by radiation during calorimetric experi- ments. Establish a relation for finding the specific heat, of liquids by the method of cooling. (Nagpur, B.'Sc.) 2. If a body takes B minutes to cool from 100°C. to 60 C C„ how long will it take to cool from 60°C> to 20*C, assuming that the temperature of the surroundings is 10°C, and that Newton's law •of cooling is obeyed. 3. Describe Joly's differential steam calorimeter and explain how it. is used for finding the specific heat of a gas at constant volume. State the corrections to be made. (Punjab Univ., 1952, 1954 ; jerat Univ., T 95 1 ; Patna Univ., 1947; Calcutta Univ., 1947.) 4. Describe the steam calorimeter. Explain how it may be letermine (i) the specific heat of a gas at constant volume, rific heat of a small solid. (A. U., 13. Sc.) 5. Describe a method of determining the specific beat of a gas at constant volume, giving a neat diagram of the arrangement of the apparatus necessary. Why is the specific heat at constant pressure ter than that at constant volume ? (Dacca, B.Sc.) 6. Describe the constant flow method of Callendar and Barnes for the measurement of the mechanical equivalent of heat. In an experiment using this method, when the rate of flow of water was i, per minute, the heating current 2 amperes and the difference of potential between the ends of the heating wire 1 volt, the rise of temperature of the water was 2-5°C, On increasing the rate of flow to 25.4 gm, per minute, the heating current 3 amperes and the potential difference between the ends of the heating wire to 1-51 volts, the rise of temperature of the water was still 2.5 °G. Deduce the value of the mechanical equivalent of heat. (A. U., B.Sc.) 7. Describe Nernst vacuum calorimeter and indicate briefly how it has been used for measuring specific heats at low temperatures. (A. U., M.Sc, 19,25,) 8. Give an account of the continuous flow method of measuring the specific heat of a gas at constant pressure and point out its advantages. (A. U., B,Sc, 1938.) 9. Describe an accurate method of measuring the specific heat of a gas at constant pressure. (Punjab Univ., 1941; Gujerat Univ., 1949, 1951.) EXAMPLES 333 '-'; 10. In a determination of the specific heat at constant pressure r Regnault's method the gas is supplied from a reservoir whose volume is 30 litres at 10°C. The pressure of the gas in the beginning is 6 atmos. and in the end 2 atmos., the temperature remaining constant at 10°C. The gas was heated to I50°'C. and led into a calorimeter at 10-0°C. The final temperature of the calorimeter and contents was 31-5°C. and its water equivalent was 210 gm. If the ly of the gas is 0*089 gm. per litre at N. T. P., calculate its. c heat at constant pressure. 11. A quantity of air at normal temperature is compressed slowly, (b) suddenly, to j\, of its volume. Find the rise of (a temperature if any; in each case [Ratio of the two specific heats of air = 1*4, log 2-75 == -4362 ; log 6-858 — 0'83fi2.] Deduce the formula used for (b) . (A. U-, B.Sc.) 12. In a Wilson apparatus for photographing the tracks of particles the temperature - of the air is 20°C. li its volume is in- creased in the ratio 1-375:1 bv the expansion, assumed adiabatic, calculate the final temperature of the air. (The ratio of specific heats of air = 1-4L) 15. Distinguish between adiabatic and isothermal changes and show that for an adiabatic change in a perfect gas jbu *= constant where y is the ratio of specific heat at constant pressure and constant volume respectively. (Allahabad Univ., 1952; Punjab Univ., 1954, 9 no.) 14. Deduce from first principles the adiabatic equation of a perfect gas. A motor car tyre is pumped up to a pressure of two atmospheres at 15°C, when it 'suddenly bursts. Calculate the resulting drop ir the temperature of air. (A. U., B.Sc, 1938.) 15. Describe a method of determining the ratio of the specific heats of a gas at constant pressure and constant volume. How has the mechanical equivalent of heat been calculated from a k of this ratio? (A. U-, B.Sc.) 16. Explain why the specific heat of a gas at constant pressure is greater than that at constant volume. Obtain an expression for the difference between the values for a perfect gas. Find the numerical value of their ratio for monatomic gases. (Punjab Univ., 1949, 1951; Utkal Univ., 1952.) 17. Describe the Clement and Dcsonncs' method of finding the ratio of specific heats of air, giving the simple theory of the method. What are the objections to the method and whnt modifications and improvements have been proposed. (A, U., 1950; Bihar Univ., 1954; Utkal Univ., 1955; Punjab Univ., 1953, 1958.) 18. Derive the relation between volume and temperature of a mass of perfect gas undergoing adiabatic compression. A quantity of dry air at 15°C is adiabatically compressed to -J-th of its volume/ Calculate the final temperature given y — 1-4 and' 4*'* = 1-74 (Aligarh Univ., 1948; Punjab Univ., 1957.) 334 F.XAMl'U'5 19. Explain how the mechanical equivalent of heat can be deduced from a knowledge of die specific heats of air at constant pressure and constant volume. State clearly any assumptions made in your reasoning and describe experiments, if' any, -which afford justification for such assumption. (Madras, B.Sc.) 20. Describe a method of determining the ratio of the two speci- fic heats of a gas. Show that it follows from the kinetic theory of gases that the ratio of the two specific heats in the case of a mona- tomic gas is 1.66, (Madras, B.Sc.) 21. Find the ratio of the specific heats of a gas from the follow- ing data : — A flask of 10 litres capacity weighs, when exhausted, 160 gm. ; filled with the gas at a pressure of 75 cm, of mercury it weighs 168 gm. The column of the gas, which when confined in a tube closed at one end and maintained at the same temperature as the gas in the flask, responds best to a fork of 223.6 vibrations per second, is 50 cm. 22. Determine the ratio of the specific heats of air from the following data :— Velocity of sound =34215 cm. per sec. in air at 750 mm, and 17°C ; density of air — ,00129 gm. per c.c, at N. T. P, ; coefficient of expansion of air— ^ £=981 cm. /sec. 2 j density of mercury e= 13.6 gm, per c.c, (Manchester, B.Sc.) Ill I. In an experiment with Joule's original apparatus the mass of the ; on either side was 20 kilograms and each fell through a one metre forty times in succession. The water equivalent calorimeter and its contents was 6 kilograms and the rise in temperature during the experiment was 0.62°C. Calculate the value of the mechanical equivalent of heat, 2. Determine Lhe heat produced in stopping by friction a fly- "' kilograms in mass and 50 cm. in radius, rotating at the rate ond assuming the fly-wheel to be a disc mounted axial b and having a uniform distribution of mass, 3. A canon ball of 100 kilograms mass is projected with a velocity of 400 metres per second, Calculate the amount of heat which would be produced if the ball were suddenly stopped. 4. In one hour a petrol engine consumes 5 kilograms of petrol whose calorific value is 10,000 cals. per gram. Assuming that h per cent of the total heat escapes with the exhaust gases and that 12 per cent of the heat is converted into mechanical energy, find the average horse-power developed by the engine and die initial rate of rise of temperature of the engine per minute. Radiation losses mav be ignored and the water equivalent of the whole engine is 40 kilograms. 5. Give the outlines of the methods bv which the mechanical equivalent of heat can be determined. Assuming- that for mi no air at constant pressure the coefficient of expansion is 1/273, the density at C. and atmospheric pressure is 0,001293, the specific heat. -fc EXAMPLES 335 ^==0.2389 and the ratio c p fe s = 1.405, calculate the mechanical equivalent o£ heat. Suppose that there is inappreciable cohesion between the molecules. (Bombay, B.Sc.) 6. Define the mechanical equivalent of heat. If the kinetic energy contained in an iron bail, having fallen from rest through 21 metres, is sufficient to raise its temperature dirough 0.5 °C V calculate a value for the mechanical equivalent of heat (given g = 980 cm. per sec. per sec. and specific heat of iron = 0J) . 7. Show how the method of electrical heating has been adopted in the determination of the mechanical equivalent of heat. One gram of water at 100°C. is converted into saturated vapour at the same temperature. Calculate the heat equivalent of the external work done during the change. Density of water at 100°C. = 0.958 gms. per c.c. ; density of saturated steam at 100°C. = 0.000598 gms. per c.c. 8. Describe a laboratory method of determining J. Give expe- rimental details. (Delhi Univ., 1951.) 9. Describe with relevant theory Rowland's method of finding the mechanical equivalent of heat. Point out the significance of the result. (Nagpur Univ., 1956.) 10. The height of the Niagara Falls is 50 metres. Calculate the difference in temperature of the water at the top and bottom of the Fall if J — 4.2 X 10 T ergs/cal. 11. A lead bullet at temperature of 47.6°C strikes against an obstacle. If the heat produced by the sudden stoppage is sufficient to melt the bullet, with what velocity the bullet srikes the obstacle ? It is assumed that all the heat goes to the bull el. Melting point of lead = 327°C, specific heat of lead = .03 i ./degree, latent heat of fusion of lead = 6 cal./gm. f J = 4.2 X 10 T ergs/cal. (U ileal Univ., 1 950,) 12.^ Describe Callendar and Barnes' continuous flow method of measuring the mechanical equivalent of heat. State how the method can be adopted to measure the variation of specific heat of water between 10 Q C and 90 C 'C. (Punjab Univ., 1951 ; A. U., 1951.) i:>. Explain what is meant by the "velocity of mean square" of molecules of a gas and their "mean free path". Show how thsse two quantities can be found. (Bombay, B.Sc.) 14. Calculate the molecular velocity (square root of the mean square velocity) in die case of a gas whose density is 1.4 gm. per litre at a pressure of 76 cm, of mercury. Density of mercury — 18.6, g=:981 cm. per sec, per sec. (Manchester, B.Sc.) 15. Show that pressure of a gas is equal to two-thirds of the kinetic energy of translation per unit volume. Calculate the kinetic energy of hydrogen per gram-molecule at 0°C. (A. U., B.Sc, 1949) . of gases this theory ? (A. U., B.Sc. Deduce Boyle's and Avogadro's laws from the kinetic theorv . What interpretation of temperature is given according to 17. Outline the essential features of the kinetic theory of gases *md an expression for the pressure of a gas on the basis of kinetic theory. (Punjab Univ., 1954, 1955, 1957; Delhi Univ., 1954.) 336 EXAMPLES 18. Show that the pressure exerted by a perfect gas is | of the kinetic energy of the molecules in a unit volume. Explain on the basis of the 'kinetic theory (i) why the temperature of a gas rises when it is compressed, and (ii) why the temperature of an evaporat- ing liquid is lower than its surroundings, (Punjab Univ. r 1956.) 19. Deduce an expression for the conductivity of a gas from the kinetic theory. How would you actually proceed to determine the conductivity of any particular' gas ? (A. U„ B.Sc, lions,, 1931.) 20. State the law of equipartition of energy. Prove that for a monatomic gas, the value of gamma, the ratio between the specific heats is 5/3 and for a diatomic gas it is 7/5. (A, IL, B.Sc., 1932,) 21. l r ind an approximate expression for the mean free path of a molecule in a gas, and give a short account of any one phenomenon depending on the length of the mean free path. (London, B.Sc., Horn.) 22. On the basis of the kinetic theory deduce an expression for the viscosity of a gas in terms of the mean free path of its mo cules. Show that it is independent of pressure but depends upon the temperature of the gas. (Baroda Univ., 1954,) ^ 23. What is meant by (a) the "coefficient of viscosity" of a gas, he ri-'. free path of its molecules"? Show how to deduce a relation between these quantities from the kinetic theory, (Lond., B.Sc.') 24. Describe some phenomena which have led to the conclusion, that molecules have a finite diameter and mean free path. How can atter be determined? IV 1. How has van der Waals modified the isothermal equation for a gas ? Calculate the values of the critical pressure, volume and tem- rature in terms of the constants of his equation.. How do the theoretically derived results tally with experiments? (A. U„ B-Sc, 193.1.,) 2. Derive van der Waals" equation of state and obtain expres- sions for the critical temperature, pressure and volume in terms of the constants of van der Waals' equation. (Punjab Univ., 1957 ; Delhi Univ., 1953 ; Bombay Univ., 1953 ; Patna Univ., 1948 ; Calcutta Univ., 1948 : Aligarh Univ.. 1950 ; Nagpur Univ., 1953 ; Baroda Univ., 1954.) 3. Express the value of the critical temperature in terms of a, b and R, Calculate its value for CO... where a — .00874 and h = .0023. (Punjab Univ., 1957 ; Allahabad Univ., 1952.) 4. Define critical temperature, pressure and volume of a vapour and give some account of the behaviour of a substance near the criti- cal point. (A. U., B.Sc.) 5. Draw a diagram showing the general form of the isothermal including both liquid and vapour state, and explain the meaning of the different parts of the curve. What is the true form of the straight portion of this curve and why ? (A, U., B.Sc, 1935.) rXAMPLES m 6. Give an account of the properties of fluids in the neighbour- hood of the critical point. Describe bow you would determine the critical constants of a substance, (A. IL, B.Sc., Hons,, 1928). 7. What is meant by the critical point in the state of a fluid ? Show on a diagram the character of typical isothermals of a fluid above and below the critical temperature. Explain how van der Waals' equation accounts for the existence of a critical point. (London, B.Sc.) 8. Explain how van der Waals' equation accounts for the exist- ence of a critical point. Calculate the values of tile critical pressure, critical volume and critical temperature for a gas obeying van der Waals' equation. (A, U., B.Sc, 1947.) 9. Describe the experiments of Andrews on carbon dioxide. State and discuss the results obtained by him. Hence show that liquid and gaseous states are only distant stages oE a long series of continuous changes. (Bombay Univ., 1953 : Baroda Univ., 1954 ; Punjab Univ., 19 10. Explain in brief outline the reasons which led van der Waals to his equation {p + afv*) (v — b) =RT. Discuss how far \i equation is in keeping with' experimental facts. (Allahabad Univ., 1950.) IL Describe Arnagat's experiments on die compressibility of gases at high and low pressures. State his important conclusion*. Indicate how the results can be explained with the help of the van der Waals' equation. (Madras Univ.) 12. Derive the reduced equation of a gas starting from van der Waals' equation of state. Show that if two gases have the same reduced pressure and volume, they also have the same reduced tem- perature, (Punjab Univ.) V 1. How would you determine the vapour pressure of a liquid above its normal boiling point ? Explain how clouds are formed by the mixing of warm moist air with cold moist air, (A. U., B.Sc) 2. Describe a method of determining the vapour density of a volatile liquid, and explain the theory of your method. (Dacca, B.Sc.) Give an account of a method which has been adopted for the determination of the pressure of saturated vapour between I00°C,. and I2(PC, Explain clearly what is meant by the statement that the specific heat of saturated vapour at 100°C is negative. /Madras B.Sc.) v 4. An electric current of 0.75 ampere is passed for 30 minutes through a coil of wire of 12.4 ohms resistance immersed in benzene maintained at its boiling point, and 29.85 gm. of benzene are found to have vaporised. Calculate the latent heat of vaporisation of benzene. 5. What are the principal differences between saturated and unsaturated vapours? How would you determine the pressure of saturated water vapour at temperatures between 40 °C and 11.0°C? 338 EXAMPLES 859 6, A mixture of a gas and a saturnLcd vapour is contained in a closed space. How will the pressure of tJie_ mixture vary (a) when the Leinperature is changed and Lhe volume is kept constant, hen the volume is changed and the temperature is kept constant ? 7, Describe and discuss a method by which you could determine the latent heat ol a metal which melts at about 20Q e C 8, Define latent heat of evaporation ol liquids and describe . how it can be measured accurately. Stale Trouton's law connecting atenl Nat of evaporation with the absolute temperature of the boiling point. VI 1. Write an essay on 'artificial production of- cold'. (A. U., B.Sc, Hons., J 030.) 2. Describe the manufacture of liquid air. (A. U., Ji.Sr,, 1030.) 3. Discuss theoretically the production of cold by expansioi gases through porous plugs. I low has iiuriple bee; in m:: or liquefying air? (A. U., B.Sc, 1937; Dell 1953, 1954, 1959.) 4. Write a short essay on the liquefaction of the so-called per- manent gases. (Dacca, B.Sc.) 5. ig hydrogen, and explain the principle involved in the process. (A. u., B.Sc.) 6. Describe and discuss the porous plug experiments of Joule and Kelvin. Explain what is meant ice to hydrogen and h< ii the meth; in the manufacture of I '.Sr., 193 8. Dii tiabatic change and Toule- Thomsoii cribe hov. ier has been utilised for I \. V., B. 7.) !i. •.: porous plug experiment of Joule and Thomson and discuss the result ed tor differed h special refer- ence to hydrogen. Indicate hov.' the results have been utilised for the Hi ►! ail and h] (Bihar Univ., 1951; Punjab Univ., 19! 10. What Is Joule-Thomson effect? Obtain the expression for iced assuming that the gas obeys van de equation. Wlr, do hydrogen and helium show a heating ordinary temperature? (Allahabad Univ., 1955; Rajasthan Univ., I960.) ' VII 1. Describe and explain a method of measuring the linear expansion of solids by means of interference bands. 2. If a crystal has a coefficient of expansion 13 X 1G"" T in one direction and of 231 X 10- T in every direction at right angles to the first, calculate its coefficient of cubical expansion. (London, B.Sc.) 3. A lump of quartz which has been fused is suspended from a quartz fibre and allowed to oscillate under the influence of the torsion of the fibre. If the coefficient of linear expansion of the material Is 7 X 10- T and the temperature coefficient of its rigidity is -f- 13 X 1®~S how many seconds a day or what fraction of a second a day would a nge of temperature of 1°C1 make? (London, B.Sc.) •1. A seconds pendulum is one which completes half an oscilla- tion in I second. Such a pendulum of invar is given and is correct at I0*C. If the average temperature for the three months of June, July and August is 25°C. and the clock is correct at 12.0 a.m. on June ]st, how much will it be incorrect at 12.0 a.m. on September 1st ? Coefficient of expansion of invar is 1 X 1Q - *** 5. Describe, in full detail, the method by which the expansion of crystals, when heated, may be studied experimentally. (Allahabad Univ., 1943; Aligarh Univ.. 1949; Punjab Univ., 1946, 1954.) 6. Describe a method by which the cubical expansion of a liquid can be accurately determined by weighing a solid of known expansion in it at two known tempi A solid is found to weigh 29.9 gms. in a liquid of specific gravity ' T:. its weight in air bei gms. It we3 in the same liquid at 25°C pecific gravity is 1.17. Calculate the coefficient of cubical ol the 7. («) Describe K method for the determination oi coefficient of absolute expansion of ,. Indicate briefly the pre- ■ led by Regnault to avoid errors. (h) If the coefficient ol cubical expansion of glass and mercury 1 and 1,8 what fraction of the whole volume of a Id be filled with mercury in order that the empty p a it should re i nsta n t when gl ass and ed to the same i mre. (Dacca, B.Sc.) S. Describe and explain the interference method For finding the coefficient : expansion of crystals. Ii 1 ss the rial expansion of a crystal differ from that of an c metal, (Punjab Univ., 1961.) 9. Describe I' s method of finding the absolute co- efficient of expansion of nn te calculations 3 in it. Why is this method better than others? (Calcutta Univ., 1949.) 1 ii The difference between the fixed points of a mercury therm o- is 18 cm. If the volume of the bulb and capillary tube up to the 0° mark is 0.0 c.c, calculate the sectional area of the capillary. ■•Jem of cubical expansion of mercury == T.S X MH*> coefficient of lii mansion of glass — 7 X JO -6 . (Madras Ur 11. An air bubble rises from the bottom of a pond, where the temperature is 7°C, to the surface 7 metres above, at which the temperature is 17°C. Find the relative diameters of the bubble in the two positions., assuming that the pressure at the pond surface is equal to that of a column of mercury of density 13.6 gm. per cc. and 76 cm. in height. 340 EXAMPLES VIII 1. By what processes does hot water in an open vessel lose heat ? Describe experiments by which the several causes of loss may be shown to exist. (Dacca, B.Sc.) 2. Define the thermal conductivity of a substance and describe some way of finding it. An iron boiler 5/8 inch in thickness exposes 60 square feet of surface to furnace and 600 lbs. of steam at atmospheric pressure are produced per hour. The thermal conductivity of iron in inch-lb.-scc. Slits is 0.0012 and the latent heat of steam is 536. Find the tem- perature of the underside of the heating surface. Explain why this is not the temperature of the furnace. (Dacca, B.Sc.) 3. Explain the difference between the thermal conductivity and the diffusivity of a substance. The two sides of a metal plate 1.5 square metres in area, and 0.4 cm. in thickness are maintained at 100°C. and 30 D C. respectively. If the thermal conductivity of the metal be 0.12 C.G.S. units find the total amount of heat that will pass from one side to the other in one hour. (Dacca, B.Sc.) 4. The interior of an iron steam-pipe, 2.5 cm. internal radius, carries steam at 140 C C. and the thickness of the wall of the pipe is 3 mm. The coefficient, of emission of the exterior surface (heat lost econd per sq. cm. per degree excess) is 0.0003 and the tempera- ture of the external air is 20 C C. If the thermal conductivity of iron is 0.17 C. G. S. unit find the temperature of the exterior surface, and bow much steam is condensed per hour per metre length of tube, a tent heat of steam at 140°C, being 509. (London, B.Sc.) 5. The thickness of the ice on a lake is 5 cm. and the tempe- of the air is - 10°C. At what rate is the thickness of the ice dmately how long will it take for the thickness ■ oubkd ? (T ■; conductivity of ice == 0.004 cal. cm- 1 sec-* °C~\ Density of ice — 0.92 gm. per c.c' Latent heat of ice = 80 cal. per gm.). 6. Distinguish between thermal conductivity and therm ometric conductivity of a substance. Describe a method of finding the thermal conductivity of a solid. Calculate the rale of increment of die thickness of ice layer on a lake when the thickness of ice is 20 cm. and the air temperature is -40°C. Thermal conductivity of ice = 0.004 cal. cm.- 1 "^C- 1 , density o£ ice = 0.92 gm./c.c. and its latent heat = 80 caL/jgffl. After what time the thickness will be doubled ? (Bihar Univ., 1954.) 7. Define the terms conductivity and diffusivity as used in the thcory of heat conduction. Describe a method of comparing the con- ductivity of two metal bars. Account for the fact that the evapora- tion of liquid air is greatly reduced when kept in a Dewar vacuum vessel. (Madras, B.Sc.) 8. Define conductivity. Deduce expression for the flow of heat in a long; bar when it has acquired a steady state. (B. H. U., B.Sc, 1931J EXAMVLliS Ml 9. Define the coefficient of heat conductivity of a substance and give details of some method of determining this constant for iron.'"' (A. V., B.Sc. 1931.) 10. Describe lngen-Hausz's experiment, and prove from the mathematical theory that the conductivities of different bars vary as the square of the length up to which wax is melted. (A. U., B.Sc, 1945, 1950.) IL Distinguish between thermal conductivity and thermometrie conductivity- Bring out, the connection between the two. Describe a tie's method of determining the thermal conductivity of a solid. (Punjab Univ., 1950.) VI. Show that in the steady state of a metal bar heated at one end Ifr = *** ■where the symbols have their usual significance. Hence prove that the length to which the wax melts in the steady state along a wax coated bar is proportional to the square root of the coefficient'of thermal conductivity of the material of the bar. (Allaha- bad Univ., 1955.) 13. Describe Forbes' method of determining- the thermal con- ductivity of a metallic bar, and explain the formulas used. (Lucknow Univ., 1950 ; Allahabad Univ., 1951 ; Punjab Univ., 1954, 1955, 1957 J Utkal Univ., 1953.) 14. Define thermal conductivity, A steady stream of water flowing at the rate of 50 grams a minute through a glass tube 30 cms. long, 1 cm. in external diameter and 8 mm. in bore, the outside of which is surrounded by steam at a pressure of 700 mm., is raised in temperature from 20°C. to S0 a C. as it passes through the tube. Find the conductivity of glass. You are given that log 1.25 = 0.223. Deduce anv formula that you use. (A, IL, B.Sc, 1937 ; Patna Univ., 1948.) 15. Define thermal conductivity. Describe a method you have adopted in experimentally finding this constant for a good conductor. Find the coefficient of conductivity of a badly conducting material upon which, the following experiment was made ;— A very thin- walled hollow silver cylinder 40 cm. in diameter and 50 cm. in length is covered all over its external surface including 1 the ends by a layer of the material 0.33 cm. in thickness. Steam at a temperature of 100°C is passed through the cylinder and the external temperature is 20°G r is found to accumulate within the cylinder at the rate of 3 gm. per minute. The latent heat of vaporization of water at 100°C. is 537 cal. per gram and cross-sections of the steam inlet and outlet are each 10 square centimetres. (Madras, B.Sc.) 16. Define conductivity and diffusivity of hcaL When steam is passed through a circular tube of length I and having the internal 342 JCXAMPIJES and external diameters a and b respectively, prove that the radial flow of heat outwards is given by 2irKl((t x -e 2 )/lag e ~ where K is conductivity and $ t , 0. 2 the temperatures inside and out- side the tube, How will you determine the conductivity of india- rubber? (A, U., B.Sc„ lions., 1930.) 17. Steam at 100°C. is passed through a rubber tube, 14.1? cm. length of which is immersed in a copper calorimeter of thermal capa- city 23 cat, containing 440 gm. of water. The temperature of the water and calorimeter is found to rise at the rate of 0.019 d C. every second when they are at the room temperature (22 °C.) . The external and the internal diameters of the tube are 1,00 cm, and 0,75 cm. respec- tively. Calculate the conductivity of india-rubber. 18. Prove Lhat if a long bar is periodically heated at one end, the law of propagation of heat is given by the equation dd _ _^0 " * iP ' Obtain a solution of this equation and show how with its aid the diurnal and annual variation oh temperature at some depth below the surface of the earth can be explained. (A, U., M.So, 1926.) 19. Heat is supplied to a slab of compressed cork 5 cm. thick and of effective area 2 sq. metres, by a heating coil spread over its ace. When the current in this coil is 1.18 amp. and the potential difl s its ends 20 vol is, the steady temperatures of the E the slab are 12.5°C. and 0°C. Assuming that the whole of the heat developed in the coil is conducted through the slab, calculate conductivity of the cork. Define coefficient of thermal conductivity and describe Lees' hod for determining the thermal conductivity of metals. Equal jer and aluminium are welded end to end and lagged. If i 1 ends of copper and aluminium are maintained at 100 o C and 0°C respectively, find the temperature of the welded interface. Assume the thermal conductivity of copper and aluminium to be 0.92 and 0.50 respectively. (Punjab Univ., 1953, 1956, I960.) 21. Describe Lees' method of determining the thermal conduc- tivity of a bad conductor, (Bombay Univ., 1947.) --. Describe and explain the cylindrical shell method of di mining She conductivity of a solid. (Bombay Univ., 1948.) 23. Describe and give the theory of a method useful for a practical determination of the thermal conductivity of a liquid. (London.. B.Sc.) 24. Discuss some methods by which the thermal conductivity of a gas has been determined. What are the experimental difficult.': and how have they been overcome ? 25. How is the thermal conductivity of a liquid determined ? EXAMPLES 3*3 How do you demonstrate that hydrogen is more conducting than air? (A. U., B.Sc, IS 20, Discuss the difficulties which beset the investigation of the thermal conductivity of gases and indicate how and to what extent they have been overcome, (London, B.Sc.) IX I. What do you understand by a reversible cycle as opposed to an irreversible one? Give instances of each, (A. U., B.Sc., 1931.) What is meant by a reversible change? Describe Carnot's cycle and prove that the efficiency of all reversible engines working between the same two temperatures depends only on the temperature of the hot and cold bodies. (Bombay Univ., 1947, 1948, 1949 ; Lkkal Univ., 1954; Punjab Univ., 1953.) 3. Describe Carnot's cycle and prove Carnot's theorem. What •nt by a reversible change ? State briefly how Carnot's theorem h to an absolute scale. (Delhi Univ.., 1952, 1954). 4. State and explain the significance of the second law of thermo- dynamics. Show that the efficiency of a reversible engine is maximum. iiv., 1950 ; Punjab Univ., 1957.) Describe a Diesel engine and deduce an expression for its efficiency. Can the Carnot engine be realised in practice? (Punjab Univ., 1949.) fi. Describe with diagrams an Otto engine and deduce an ex- ion for its efficiency. (A. U., B.Sc, 1949.) 7. Describe some kind of internal combustion engine and explain fully how heat is thereby converted into work. Describe its uses and applications and dwell upon its advantages over the steam engine. (A. U., B.Sc.) 8. Describe the cyclical process of a steam engine and compare its efficiency with that of an internal combustion engine. Explain (1) why steam engine is preferred in railways, and (2) why petrol engine is used in. aeroplanes. (A. U,, B.Sc, 1930.) 9. Write an essay on 'Heat engines'. (Dacca, B.Sc) 10. Explain the indicator diagram and apply it to Carnot's cycle. Explain the conditions for reversible working and show that in ge for reversible cycles 1 -y=0. (Bombay, B.Sc) I I . Describe Carnot's cycle. Show how the work done during each operation is represented on a pv diagram. Find an expression for the Work done during ouch operation when the working substance is a perfect gas, (London, B.Sc.) 12. Discuss the statement : 'Reversibility is the criterion of perfection in a heat engine'. Explain with two examples what vou understand by a reversible process. A Carnot engine works between the two temperatures I00°C. and 10*C. Calculate its efficiency. (A. U., B.Sc, 1947.) 344 EXAM X EXAMPLES 345 1. Show that when a body expands, the external work performed is given by the expression From the following data calculate what fraction of the specific heat of copper is due to the external work done in expansion in an atmosphere at a pressure of 76 cms. of mercurv :— Specific heat of copper = 0.093, Specific gravity of copper = 8,8, of mercury = 13. 6. Coefficient of linear expansion of copper = 0.00 On 1 6, J = 4,2 X 10 7 ergs per calorie. (A. U., B.Sc) 2. Deduce an expression for the work required to compress adiabatically a mass of gas initially at volume v x and pressure pi to volume v 2 . Find the work required to compress adiabatically I gm. of air initially at N.T.P. to half its volume, Densitv of air at N.T.P, === 0.00129 gm./c.c. and c p jc =z 1.4. (Birm,, B.Sc.) 3. Explain external and interna! work. Discuss the changes in the kinetic energy of the molecules of a gas when heated and hence show that the ratio of the specific heats for a monatomic eai is 5/3. * 4. Write an essay on the transformation into heat of other forms of energy. (Cal, B.Sc.) 5. Tn what sense can the second law of thermodynamics be ded as furnishing ail absolute scale of temperature ? How can readings of gas therm i be reduced to this particular scale ? !J.; B.Sc, Bom., 1931.) -i what you .i id by a thermodynamic scale of rature. Show that it agrees math an ideal uas scale. (Punjab Univ., 1053, 1954, 1956, 1957; Bombay Univ., 1953.) 7. Write notes on the following : — (a) Lord Kelvin's absolute scale of temperature, (b) Joule and Kelvin's porous plug experiment, ' B.Sc, 1930.) 8. How did Kelvin arrive at the absolute scale of temp Show that the ideal gas scale and the absolute scale are Iden- [Iow is the absolute scale I in practice ? (Delhi Univ., (Nagpur, ture ? deal. 1951, 1953, 1956.) 9. Define a scale oF temperature without malting use of the peculiarities of any selected thermometrit substance. Show (a) that Kelvin's work scale is such a scale, and (b) that the ratio of two tem- peratures as measured on the Kelvin scale is identical with the ratio of the same two temperatures on the perfect gas scale. (London, 10. Explain what you mean by die entropy of a substance. Show that for any reversible cyclic change of a system the total change of entropy is zero. Explain why this statement is not true for irrever- j changes. (Punjab Univ./ 1958; Baroda Univ., 1955.) 11. Explain the idea of entropy. Derive an expression for the entropy of m grams of perfect gas. (Allahabad Univ., 1950.) 12. A volume of a gas expands isotherm ally to four times its initial volume. Calculate" the change in its entropy in terms of the gas constant. (Baroda Univ., 195C) 13. Calculate the change of entropy when 100 gms. of water at S0 Q C. are mixed with 200 gms. of water at 0° assuming that the specific heat of water is constant between these temperatures. (London, B.Sc.) 14. Derive an expression for the entropy of a perfect gas in terms of its pressure, volume and specific heats. (Bombay, B.Sc.) 15* Prove that the increase of entropy per unit increase of volume under constant pressure is equal to the increase of pressure per unit increase of temperature during an adiabatic change. (Bombay, B.Sc.) 16. Calculate the change of entropy when 10 gm, of steam at 100°C, cools to water at 0°, assuming that the latent heat of vaporisation is 536 and the specific heat of" water is 1 at all tempera- tun. 17. State the second law of thermodynamics and apply it to the determination of the effect of pressure on the melting point of a solid. (A, U., B.Sc) 15. Give an elementary proof of Clapeyron's relation dp L dT 1 (r s — v L Y Discuss how the boiling- point of a liquid and the melting point ■a solid are affected by change of pressure. (Punjab Univ., 1917, 1957.) 19. Derive Clapeyron's equation dp L _ dT 7{p 2 — v t ) ' Calculate the change in temperature of the boiling point of water due to a change of pressure of 1 cm. of mercury. (L = 536 calories, volume of 1 gm. of water at 100°C = 1 c.c, volume of 1 gm. or saturated steam at 100°C — 1600 c.c.) . (Delhi Univ., 1955.) 20. Prove the thermodynamic relation >a\ tdp\ \ dv Id ' \dOL and hence prove ft'0 (*-*) (B. H. U„ B.Sc, 1931.) 346 KXAMPi.KS 21. Deduce the latent heat equation Calculate the depression of the melting point of ice per atmos- pheric increase oL" pressure, given latent heat of fusion — bO cal. and njity of ice at 0°C. 10 ~, „ c ir iar.\ r nor -, — rr- (Nagpur, B,Sc„ 1930.) density of water at O'C. 11 v bl l 22. Prove that One gram of water-vapour at 100°C, and atmospheric pressure occu- pies a volume of 1640 c.c. and L — 53G calories, Prove that the vapour pressure of water at 99°C. is about 733 mm, of mercury. (A, U., RlSc, 1926.) 23. If sulphur has a specific gravity 2.05 just before, and 1,95 just after melting, the melting point being 115°G. and the latent heat 9.3, find the alteration in melting point per atmospheric eha of pressure. (I.ond., B.Sc, Hons.) 24. Calculate the von I me of n gramme of steam at 100°C, given that the lati i of evaporation of water at 10G°C = 535 and the change of boiling point is 0,37°G. per cm. of mercury pressure (/ £=4.2X10*). (Manch., B.5c.) 25. Calculate £ given that the change of of one phere changes the melting point of ice by C and when one gram of ice melts volume changes by Q.O907 (Punjab t 26. Discuss the effect of change of pressure on the polling point of a liquid. olum.es of water and saturated steam at 1Q0°C are 1 c.c. and 1601 c.c. respectively and the latent heat vaporisation is 536 cal./gm., find the change in boiling point for a change of pressure of 1 cm. of mercury. (Allahabad Univ., 1946 j Calcutta Univ., 1948.) 27. Show, on thermodynamic principles, that when a liquid film is suddenly expanded, it must fall in temperature, and find an expression for the lieat that must be supplied to keep it constant. (A. U., B.Sc, Hons., 1927.) 28. Deduce an expression for the specific heat oi saturated vapour and prove that in the case of water at 100°C., It is negative. How do you explain the paradox 'negative specific heat ?' (A. U., B.Sc., Hons., 1931.) 29. Define the 'triple point*. Describe the successive changes observed in a system containing water at its various states when pressure is changed at a constant temperature (a) when the tern- ■e is above the triple point, (b) when the temperature is bel< it. (Dacca. B.Sc., 1930.) EXAMPLES 30. The latent heat of steam at 100°C. is 536, calculate what fraction of this heat is used up in performing external work during vaporisation, assuming the density of steam at 100°C to be 0,007 and an atmosphere to be 10 c dynes per sq. cm. (Man eh-, B.Sc.) 31. Define the 'triple point' and show that the steam line, hoar- frost line and the ice-line must meet in a single point. Draw the isodiennals of water for temperatures lower than that of the triple point. (A. U., B.Sc, 1934.) $2. Define Entropy. What is its physical significance? Show that the entropy remains constant in a reversible process but In- creases in an irreversible one. (A. U., B.Sc., 1936.) S3. Define Entropy. Prove from the principles of thermody- namics that the decrease of entropy per unit increase of pressure during an isothermal transformation is equal to the increase of volume per unit increase of temperature under constant pressure. Hence show that heat is generated when a substance, which expands on heating, is compressed (A. U., B<5c, 1938.) 34. What do you understand by 'entropy' ? State the second law of thermodynamics in terms of entropy. Calculate the change in entropy when m grams of a liquid of specific heat s are heated from 2V to r 2 °, and" then converted into its vapour without raising its temperature. [Latent heat at 7V =L.] (A. IP, B.Sc,, 1941.) 35. Prove the latent equation dL L dT T ~ H H Calculate the specific heat of saturated steam from the above equa- tion and explain the meaning of its negative: value. Given L = 539.3 cal., T = 100°C ; d h, = - 0.640, c t — 1.01. (Allahabad Univ., 1955 ; at Baroda Univ., 1953 ; Nagpur Univ., 1956.) XT 1. State Newton's law of cooling. A copper calorimeter weigh- ing 15 gms. is filled first with water and then with a liquid. The times taken in the two cases to cool from 65°C to 60°C. are 1 70 sec. and 150 sec. respectively. The weight of the water is 11 gms. and that of the liquid is 13 gms. Calculate the specific heat, of the liquid. The specific heat of copper ia 0.1. (Cal., B.Sc.) 2. State Stefan's law and discuss in the light of the same (I) Newton's law of cooling, (2) the temperature of the sun, and (3) the perature of a tungsten arc. (A. U., B.Sc, 1932.) 3. State and deduce KirchhofFs- law on the emission and absorp- tion of thermal radiation. (Sheff,, B.Sc.) 4. What is a black body ? What are the characteristics of a 348 EXAMPLES blackbody radiation? How has it been realised In practice? Des- cribe how Stefan's law has been verified w T ich it. (Allahabad Univ., 1949; Punjab Univ., 1961.) 5. How can you show that the radiation from an enclosure depends only on the temperature and not on the materials of its walls. Describe a radiation pyrometer, (Delhi Univ., 1955.) 6. Explain the terms 'emissive power" and 'absorptive power'. Deduce that at any temperature the ratio of the 'emissive power to the absorptive power 'of a substance is constant and is equal to the emissive power of a perfectly black body. (Baroda Univ., 1951.) 7. Discuss the evidence both theoretical and experimental show* ing that good emitters are good absorbers. (Punjab Univ., 1953.) 8. Explain what you understand by a black body. State Stefan's law of radiation and prove, it from thermodynamkal consi- deration. Indicate how it can be verified. (Punjab Univ., 1944, 1950; Roorkee Univ., 1951, 1959.) 9. Explain what is meant by a "perfectly black body". Write a short account of the distribution of energy in the spectrum of the radiation from such a body. (Leeds, BJSe.) 10. Describe any furnace and explain how you would measure its temperature, (A. U„ B.Sc) 11. Discuss the principles underlying the measurement of tern- inrii-nT pyrometers, and show how the value of the temperature is estimated. Describe fully and clearly a prac- tical form of apparatus of this type. (A. U., Br.Sc, lions., 1930.) 12. DeOne solar constant. Explain with necessary theory how the solar constant, is determined. How is the temperature of the estimated from the data of the solar constant ? (Allahabad Univ., i Punjab Univ., 1952; Nagpur Univ., 1953.) 13. Under what conditions may a thermodynamic investigation of the radiation within an enclosure be carried out? Show that under these conditions the radiation varies as the fourth power of the temperature, (A. U„ B.Sc*, Hons,, 1928.) 14. Describe the construction of Boys' radio-micrometer and explain the principles involved in its action. (London, B,Sc) 15. Calculate the temperature of the Earth, assuming that it absorbs half the energy falling on it from the sun and that the sun radiates as a black body, (Radius of Sun -7x 10 10 cm ; radius of Earth — 0,3 >( 10 s cm; mean radius of orbit of Earth = 1,5 X 1& 18 cm J temperature of sun's surface— 6Q00°K; SteFan's constant <rz= 5,7 X 10 -8 C.G.S. units,) 16. State Stefan-Boltzmann law of radiation and describe briefly experiments by which it has been confirmed. (A. U., B,Sc„ 1934.) EXAMPLES 349 (in) (y) (vi) (vii) 17. What is a perfectly black body ? How can such a body be realised in practice ? How will you verify experimentally that the ratio of the emissive and absorptive powers k the same for all bodies and is equal to the emissive power of a perfectly black body? (A. U., B.Sc, 1940.) 18. State Newton's law of cooling mentioning its limitations. How would you verify it experimentally ? \ body initially at 8D°C cools to 64°C in 5 minutes and to 52°< in 20 minutes. What will be its temperature after 16 minutes and what is the temperature of its surroundings? (Punjab Univ., 19oI, 1952 ; Patna Univ., 1951,) 19. Describe the radiation method of measuring high tempe- ratures. (Delhi Univ., 1951.) 20. Write short notes on the following :— (i) Prevost's theory ot exchange. (Punjab Univ., 195 1.) , Perfectly black body and its actual realisation. (Delhi Univ., 1952,) Blackbody radiation. (Delhi Univ., 195G.) Stefan's law. (Delhi Univ., 1955,) Solar constant. (Delhi Univ., 1956.) Determination of the temperature of the sun, (Delhi Univ., 1952.) Optical pyrometers. (Allahabad Univ.) XII 1. What is meant by (a) relative humidity, (b) dew-point ? Describe the Rcgnault's hygrometer and explain how relative humidity can be measured with its help. 2, Explain the general principles underlying the use of a wet and dry bulb hygrometer for determining the hygrometric state of the atmosphere. 3. Describe the hair hygrometer and state its uses. 4, Assuming convective equilibrium of the troposphere, find an expression for the decrease in temperature as we go upwards in the atmosphere. ANSWERS TO NUMERICAL EXAMPLES 2, 50°C. I 3. 1.55:1. 4. 1.29 X 10» ei j II 2. 8-22 min. nearly, G. 4.217 X ™ 7 ergs/cal 9. $.89 cal. per gm. per °C. 10. (a) ; (b) 412.8°C, 11. -)5.9°C. 13. 51,8°C 17. 228°C, 20. 1.60. 21. 1.402 350 EXAMPLES III 1. 4.22 >< 10 T ergs. 3. 1.91 X 10° cais. 5. 4,16 x 10 T ergs. 7. 40.5 cat, II. 3.47 X 10* cro./sec, 15. 5.67X I" 10 ergs. 2, 29,5 cal. i 9.35 h.p. ; 13.1 °C. per min. 6. 4.116 X 10 7 ergs. 10. 0.119°C. 14. 4.66 X 10* cm./sec. IV 3. 33,6 °C. {a arid h are given in atmospheres and c.c. respec- tively for 1 ex. of gas at N.T.P.). 4. 100.5 cal. per gm. V VII 2. 475X10-*. t. 59.6 sec. 10. 0.000801 sq. cm. 3. 5.646 sec. per day gain per °C. rise, 6. 0.8 X 10- 4 per °C. 7. ■ JL II. 1.20:1 VIII 3. 11.3 X 1 8 cal. 0.39 cm. per hour; 19 hours 40 minutes. 14. 1.31 X 10- 3 * 15. 1.26 X 10 B C.G.S. units, 3.54 X 10~*. 1 !i, I . ] 3 x 10-* C.G.S. units, 20. 64.8°C X i. 6.28 X 10 s ergs. 12, 1.385JJ. 13. 0.36 unite, 21. O.OOSPC, 23. 0.0252°C. 24. 1674 c.c 25. 80,8 calories. 26, 0.36°C, 30. 35. -1.07. 1 19. 0.36* XI 1. 0.733. 15. 290°K, 18. 43°C : 16°a The Gas Constant Coefficient of expansion for perfect gas at 0°C. Ice point Volume of one gram-molecule at N, T. P. = Avogadro number N = Loschmidt number n = Mass of H 1 atom M = Standard gravity g s= Density of mercury at N.T.P. = Standard atmosphere s= Mechanical equivalent of I Boltzmann's constant k = Stefan-Boltzmann constant a— Wien's constant b == Planck's constant k := Velocity of light in vacuum Electronic charge <■ ==r Faraday number Mass of the electron m ific electronic charge e/m Gravitation constant PHYSICAL CONSTANTS* R = (8.31436 ±0.00038) X 10 T ergs. deg- 1 mole- 1 = 1.98646 =h 0.00021 cal. deg- 1 mole— 1 . 0.0036608 per °C. ■273.16 ihO.OPK. ; 22.414 litres : (6.0228 ± 0,0011) X 10 28 mole- 1 (2.6870 =fc 0.0005) X 10 ,ft cm,-'' (I.67S39 ± 0.00031) X 10" 2 ' 1 gm* 980.665 cm. sec.- 2 13,59504 ± 0.00005 gm. cm-* 1,013246 X 10* dynes cm.- 2 ,i 04) X ™ 7 ergs cal.- 1 (1.38047 ± 0.00026) X 10-" ergs. (5.672 ± 0.003) X 10" 6 erg. cm- 2 deg.-" 1 sec. -1 0.28971 ± 0.00007 cm. degree (6.624 ±. 0.002) x 10 ~ 2V erg, sec. (2,99776 ± 0.00004) X 10 10 cm. c — (2.99776 ± 0.0010) X *0- 1( > e.s.u. = (1.60203 =b 0.00034) X 10~ 20 e.m.u. V = 96501 dr 10 int. coul. per gm. equiv. (9.1066 ± 0.0032) X 1() ~ 2B gm. ) xlO 7 e, m. n. per gm. G = (6.670 ± 0.005) X lO -8 dyne cm. 2 em. -3 * The values given here are taken from Birue, Reviews of Modern s, Vol. 13, p. 233 (1941) ; Amer. Jour! Phys., Vol. IS, p. 63 5). All quantities in this table involving the mole or the gram equivalent are on the chemical scale of atomic weight (O = 16.0000) . 351 / SUBJECT INDEX. The numbers refer to Pages Absolute expansion of liquids, 168 ilute null point, see Absolute zero, lute pyrheliomcter, 312 Absolute scale of tempera ture, 249 Absolute zero, 5, 2S0 Absorption coefficient, 286 Absorption freezing machines, 127, 131 Absorption of radiation, 28fi, 289 Absorptive power, 286, 289 :.Ljc change of humid air, 325 Adiabatic demagnetisation, 147 — equilibrium, 318 Adiabatic expansion method ing % 53 Adiabatic expansion of compressed gases, 132 Adiabatic stretching of wires, 265 Adiabatic transformations, 4R Adsorption, 125, 133 Air ci :- 151 et . — machine, 154 151 liquefacti ictton air. mtial, 2?8 1 1:j its, 97 machines, 131 machines, Amor lids, 105 Andrews' experiments ■ ctivity of gases, 198 Andrews' experiments on carbon dioxide, 88 s experiments, 187 isotropic bodies, thermal expansion of, 165 application of Klrcbhoff's law to, 291 Athermancy, 288 Atmospberej thermodynamics of, 316 pi seq. — , distribution of temperature in, 316, 319 — , distribution of pressure in, 318 — t water vapour in, 319 ispherfc engine, 206 Atomic energy, 79 Atomic heat, 43 — , variation with temperature, 45 Available energy, 243, Ige velocity, 78 Avogaiiro's law, 75 — number, 76 B Bartoli's proof of radiation pressure,, 301 Baths, fixed temperature, 11 — . sulphur, 12 Bcckmann's thermometer, 3 Becquerel effect, 21 Bell-Coleman refrigerator, 133 Berthelot's apparatus, 118 Bimetallic thermo-rcgulator, 174 Blackbody, — , absorptive power of, 286, 290 — , definition of, 286 — , : i —wer of, 290 — of Fery, 297 — oi Wien, 296 — , spectrum, emitted by, 305 — , total radiation from, 302 Blackbody curves, 305 Blackbody radiation, 295, 302 Blackbody temperature, 308 ■ — , measurement of, 307 et seq. — of tlie sun, 314 point of water, 11 — , variation with pressure of, 263 Bolometer, 299 — , linear 299 — , surface, 299 Bo nb calorimeter, 62 Boyle's law, 5 — , deduction from kinetic theory, 75 — >, deviations from, 87 Bridges for resistance thermometer, 15 Broadcasting waves, 284 Brownian movement, 71 Buusen'iH calorimeter, 33 Caikndar and Griffiths' bridge, 16 Callendar's continuous flow calorimeter, 38 Caloric theory, 64 Calorie, definition Calorimeter, — , bomb, 62 — , Bunseii's see, 33 — j continuous flow, 38 — j copper block, 31 — , differential steam, 36 — .i July's steam, 34 — , Nernst vacuum, 41 — , steady-flow, ■■'.. Calfcrimetry, Chapter II — , electrical methods, 37 et seq, — , method oi cooling, 32 — , method of mixtures, 29 — , methods based on change of state, 33 Carbon dioxide, critical constants of, 90, 102 — , isothermal curves of, 89 — , theoretical curves for, 93 Caruot's cycle, 213 *f seq. i hot's engine, 213 et seq. . reversibility of, 218 ; theorem, 218 Cascade process of refrigeration, 135 Change of state, Chapter V — , application of thermodynamics to, 262, 272 Charles' law, 5, 176 Chemical thermometers, 3 Claude's air liqucfier, 143 Claude-Heylandt system, 144 ,. Clausius-Oapeyron equation, 262 — , Clapeyron's deduction of, 272 Clement and Desormes* apparatus, 53 leal thermometer, 3 Coefficient of performance, 235 Combustion engines, see Internal com- bustion engines Comfort chart, 151 Comfort zone, 152 Comparator method, 159 Compensation of clocks and watches, 172 Compensation of mercury pendulum, 171 Compound engine, 212 idensation, methods of causing, 324 Conduction of heat, Chapter VIII — , in three dimensions, 191 — r Kinetic Theory of, 84 — through composite walls, 190 Conductivity, thermal, definition of, 179 — . of different kinds of matter, 178 23 , relation between electrical tivity and, 190 Conductivity of Earth's crust, 188 Conductivity of gases, S4, 197 ei seq. — , relation between, viscosity and, 84 — , variation with pressure, 84, 199 Conductivity of glass, 194 Conductivity (thermal) of liquids, 196 et seq. Conductivity (thermal) of metals, determination, 180 et seq. — , by a combination of steady and variable flow, 186 — from calorimetric measurement, 180 — from periodic flow of heat, 187 — from temperature measurements, 182 Conductivity of poorly conducting- solids, 192 et seq. Conductivity of rubber, 194 Conductivity, thermometric, 182 Constant, Boltzmann, 76 — , critical, see Critical constants — , gas, 6, 10 — , Planck's, 304, 351 — , solar, 311 — t Stefan's, 302 — , Wien's, 304, 306, 351 Constant flow method for gases, SI — for liquids, 38 Constants in kinetic theory, 85 Continuity of spectrum, 281 Continuity of states, 90 Convection of heat, 201 — , forced, 203 — , natural, 202 Convective equilibrium, 318 Cooling by adiabation expansion, 132 — due to desorption, 133 — due to Joule-Thomson effect, 136 et seq. — due to Peltier effect, 133 — , Ncwton'slaw of, 287 — , regenerative, 141 Correction for emergent column, 171 Correction of barometric reading, 170 Correction of gas thermometer from Joule-Thomson effect, 269 Crank, 210 Critical coefficients, 96 Critical constants, 90 — , deduction from van der Waals' ■:viriti;,n, •'.!.'< — , determination of, from law of recti- linear diameters, 100 — , tabic of, 102 Critical density, 101 subject iNnnx 355 Critical point, 90 — ., matter near the, 102 Critical pi i 95, 100 — temperature, 90, 95, 100 — volume, 90, 95, 100 CryophortiK, 126 Cryustatg, 149 Crystals, expansion of, 165 Cycle, Carfiot, 214 — — , efficiency of, 216 —j reversibility of I — , Diesel, 226 — »— , efficiency of, 227 — Otto, 223 , efficiency of, 224 —, Rankine's, 220 , efficiency of, 221 Cylindrical shell method o| finding CI iiiuCtivLLy, 193 I J Dalton's law of partial pressures, 75 ■ ty lamp, Dead space correction, 9 .7'' ■,'apour, m Vapour density • from I ! ■;■ Ic's law, — sei Die : air ili: • . 278 Diffuse radiati n, energy density of, 301 — , pressure of, 301 meter method, lci7 Disappearii ient pyrometer, 310 : ir poor conductors, 194 tti energy, 245 Displacement law, Wien's, 304 Distribution of energy in blackbody rum, Distril (Maxwell . ting engine, 208 ie's theory of conduction, 191 D-slide valve, 209 Duiong and Fctit's law, 43, 81 — , from kinetic stand-paint, 81 — , illustration of, -ii Dynamic method of finding vapour pressure, 114 Earth, temperature inside the, 188 Earth's crust, conductivity of, 188 Eccentric, 21 1 Effective temperature, 153 Efficiency of engines, 212 — , Camot's cycle, 216 — , Diesel cycle, — , Otto cycle, 225 l\]i.i.:trical methods in Calorimetry, 37 et seq. nprir "magnetic waves 283*234 Emission coefficient, 290 — , total, 302 Emissive power, 290 Energy, conservation of, 242 — , chscviiuiHHMK changes in, 245 — , dissipation of, 343 — , distribution of, in the spectrum, see Distribution of energy in the black- body spectrum — , forms of, 241 — , molecular and atomic, 79 — , transmutation of. 242 Engine, Carnot's, 213 — , Diesel, 226 — , hot-bulb, 228 — , Internal combustion, see internal combustion en — , jet, 234 — , National gas, 228 — , OtitO, 223 — i re ; irreversible, 218 — , semi-Diesel or hot-bull i. , single-acting, 209 Entropy, 25 1 et seq. rgy, 256 — , change of, in reversible pp ■ 253 — — , in irreversible pr icesse . 254 i definition of, 251 — , law of increase of, 255 — .if a perfect gas, 2~? — of u system, 253 — of steam, 258 — , physical concept of, 257 — -, statement of sec i thermo- dynamics, ill t< rn - of, Entropy-temperature dlagramj 257 3 T .i|i::iiii:n of Clausius-Clapeyron. 262, — of heat conduction (Fourier), 183 Equati ms of state i p, IV — , defmitirn of, 87 Kperimental study of, 96 — of van der Waals, 'Jl Equilibrium, adiabatic or convective, 318 — , radiative, 317 Equipartkion of energy, 79 Eutcctic mixture, 126 — temperature. Examples on Thermodynamics, 271 Exchanges, Frevosfa theory of, 286 Expansion (thermal). Chap. VI f — , applications of, 172-174 Expansion of anisotropic bodies, 165 Expansion of crystals, 161, 165 — j, Fizeau's method, 161 — , fringe-width dilatometer methodj 163 Expansion of gases, 175 — , determination of, volume co- efficient of. 175 Expansion of invar, 165 i 1 -. | ' •; i.i of liquids, 166 et seq, — , absolute, by hydrostatic balance method, 169 — , dilatometer method, 167 msion of silica, 165 Expansion (linear) of solids (isotro- pic), 157 — , discussion ::i results, 164 — , earlier measurements of, 158 — , measurement of, 159 cl seq. , comparator method, 159 ■, J .;•:■-■ titer and Laplace's method, 159 — relative, hy Helming r S tube method, 160 Expansion of water, 171 Expansion, surface and volume, lb5 P Film method of determining conducti- vity of liquids, 196 — of gases, 198 First law of thermodynamics, see Ther- modynamics, first law of Fixed points, chart of, 25-26 Fixed temperature baths, 11 Flow of heat (rectilinear), 182 — , combination of steady and variable, 18! i — , periodic, 187 ■heel, 210 ies' method, 186 unhofer lines, 291 — spectrum of the sun, 291 Freedom, degrees of, 79 Free path, 81 — , mean, see Mean free path Freezing mixture, 126 Freezing point, see Melting point Frigidairc, 130 Fuel in engines, 229 Fusion, 104 ei seq. — , effect of pressure on, 109 — — , thermodynamic explanation of, 262 — , latent heat of, see Latent heat of fusion - of alloys, 110 Gamma rays, 285 Gas engine-. 332 1 1 seq. Gas laws. 5 — , deduction of, from kinetic theory, 7B — , deviation of gases from, 87 et seq. perfect, 9, 87 Gas scale corrections from Joule- Thomson effect, 269 Gas scale, perfect, 9 Gas thermometers, 4 — , Calendar's compensated air thermo- meter, 6 — , constant pressure, 6 — , constant volume, 6, 7 — , constant volume hydrogen, 7 — , standard, 7 Gases, conductivity of. 1 r 7 — , equation of state for, Chap, IV — , expansion of, Chap. I & VII — , liquefaction of, 133 et seq. Gases, permanent, 194 Gases, specific heat of, 49 et seq. See also Specific heat of gases — , thermal conductivity of, 84, 197 ■ — , viscosity of, 83 Glacier motion, 110 j, conductivity of, 194 Governor, 210 Gruneisen's law, 165 Guard ring, 181 H Hair hygrometer, 323 Hampson's air liquefier, 143 Heat, a kind of motion, 64, 70 — and light. 281 — and work, 65, 243 — as motion of molecules, 70 — balance in human 81 BJECT 1NIHCX Heat by friction, 6J Heat, caloric theory of, 64 — , convection of, 201 — , dynamical equivalent of, 6S-70 — engines, Chap, IX, see also Hngines and Cycles — , latent, 104, 107, 117 — — , variation with temperature, 120 — , nature of, 64 c! scq. — of combustion, 62 — , periodic flow of, 187 at teq, — , propagation of, J 78 — , radiant, Chap, XI, see also Radiation. — , rectilinear flow of, 182 — , specific, see Specific heat — , steady flow of, 183 — t unit of, 28 Helium, liquefaction of, 146 — , solidification of, 147 Henning's tube method, 100 Hertzian waves, 282 Hess's law, 246 Hot-bulb engine, 228 JlnL-wire method of finding conducti- vity, 197, m Human body, heat balance in, 63 Humid air, adiabatic change of Humidity, 320 — , absolute, 320 — , relative, 320 Hydrogen, conductivity of, 198 — , critical temperature — , Joule-Thomson cooling in, 144, 268 faction of, 144 — spectrum, 292 i crmometer, 7 Hydrostatic balancv 169 Hygrometer, 321 — , chemical, 321 — , dew-point, 322 — , hair, 323 — , Regnauit's dew-point, — , wet and dry bulb, 3 J 3 Hygromctry, 32] Hypsometer, 12 Ice, latent heat of, 105, 107 Indicator diagram, 213 Infra-red ray?, 282 Ingen-Hausz's experiment, 184 Integrating factor, 240, 260 Internal combustion engines, 222 el seq. -, application of, 229 — , fuel used in, 229 • i i * a gas h panding, 47 i Ltional temperature cale, 24 Intrinsic energy, vnrin: , with volume, 266 Invar, expansion of, 165 Inversion temperature, I — , expression for, from thermo- dynamics, 269 Irreversible engines, 218 — process, 216 Isothermais, 89, 98 J J, ili.:'.i.:niiin;-ilii..i:-. of, 66 et scq. Jena glass, 2 Jet propulsion, 234 Joly's steam calorimeter, 34 — , differential form, 36 Joule's experiments. 47, 65 — law, 47 — method of finding y, 54 Joule-Thomson effect, 336. 268 — , Correction of gas thermometers from, 269 — for gas obeying van der Waals' — inversion of, 145 K Kapitza's tiquefier, 144, 147 Kinetic theory of matter, Chap, II I — , constants, table of, 85 — , deduction of gas laws from, 75 — , evidence of the molecular agitation, 71 — , growth of, 70 — , introduction of temperature into, 76 — of specific heat, 79-81 — , pressure of perfect gas from, 72 Kirchhofl's explanation of FraunhofLr lines, 292 Kirchhoff's law, 290 — , application to Astrophysics, 291 . deduction of, 2<>5 Langlcy's bolometer, 299 Laplace's formula, 3 If! Lapse-rale, 316 Latent heat of fusion, 104 — , determination of, 107 et seq. 247 ' m Lit. d of finding conductivity of ii !■;, 1% • ii poor conductors, • I h , 278 ion by heating, 281 l ,in 1 1 ■'■■ ii achine, 142 i [near ton, 157 et scq. ion of air, 142 Liquefaction of gases, 133, el — by application of Joule -Thomson effect, 136 et seq - by application of pressure and low temperature, 133 — by cascades or scries refrigeration, 135 Liquefaction of helium by IC O lines, 146 Liquefaction of hydrogen, 144 Liquid air, uses of, 149 Liquids, expansion of, see Expansion of liquids — , thermal conductivity of, tit Con- ivity of li Liquid thermometers, 1-4, 23 of planetary atmosphere, 78 Low temperature siphons, 149 Low temperature techniques, 148 Low temperature thermometry, 22 M Matter, continuity of liquid mid gaseous states, 90 — , state of, near the critical point, 102 Hatter, three states of, 104 Maximum and minimum thermometer, 3 Maxwell's demon, 256 Maxwell's law, graphical rcpreseuta- on of, 77 — of distribution of velocity, 76 Maxwell's tbermodynamical relation- ships, 260 et seq. Mayer's hypothesis, 47 Mean free path, 81 — of Maxwell, 82 _ Mean square velocity, 78 Mean velocity, 78 Median ir tifvalent of heat, 6fi et seq. — , value of, 70 Melting, d electrical resistance on, 106 — , change of vapour pressure on, 106 — , chat • volume on, 106 Melting point, 106 — of ice, effect of pressure on, 109, 262 — of metals, 106 , effect of pressure on, 263 Mercury, conductivity of, 181 — , expansion of, 170 Mercury thermometer — , errors oi, , correction for, Method, electrical, of measuring specific heat, 37 et seq, — of finding latent heat, 108 — of cooling, 32 — of melting — of mixtures, 29, 49 ant! latent heat of fusi' 107 — — and latent heat of fusion of metals, 108 Methods of causing condensation. Molecular I .mi, 44, 79 Molecules, diameter of, 81 — , free path of, see Mean free path — , translation;)! energy of, in gas, 76 — > velocity of, in pas, see Velod Multiple expansion engine, 212 K National gas engine, 228 Natur> :, 64 Ner nst's copper block calorimeter, 31 Mernst's vacuum calorimeter, 41 Neumann's law of molecular heats, 44 Newton's law of cooling, 29, 287 Nozzle, expansion through, 230 Optical method of measuring expan- sion, 161-164 Optical pyromctry, 309 see also Radia- tion pyrometry Otto cycle, 223 Otto engine, 223, 229 Path, see Mean free path Peltier effect, 21 — , cooling due to. 133 Pendulum, gridiron, 173 — , mercury, 171 $58 SUBJECT INDEX Perfect differential, 239 Perfect gas, 5, 87 — t pressure of, 72 — scale, 9 Periodic flow of li.-;it, 187 Permanent gases, 134 — , liquefaction of, see Liquefaction nf Perpetual motion — of first kind, 219, — of second kind, 219, 248 Fetrolei m ether thermometer, 23 Phenomena, mean free path, 81 — , transport, S2 Phenomenon of conduction. 178 (Chan. VIII) — of viscosity, S3 Photosphere, 314 — , temperature of, 314 Planck's radiation formula, 304 Planet, artificial, 23S Planetary atmosphere, loss of, 78 Platinum resistance thermometer, 13-18 Poro. :periment, 139 Potentiometer, 20 p. 97 Pre::. ' 7n 30© see ure Priji Principle of regenerative 141 iPyrhel I lute, 312 — , v, — , wi tir, 312 Pyrometry, 306 si — » fPs pyrometers, 306 — , optical, 309, see also Radiation pyp i i — , radiation, 307 — , resistance, 307 — , thermo-electric, 307 Quantity of heat, 28 Quartz, 165 R iant energy, identity of light and, 281 e! seq. — , nature of, 279 — , passage through matter of, 28fi, 288 — , properties of, 279 Radiatioi . I . a XI — , application of thermodynamics to, see Therm- dynamics of radiation — , blackbody, 293-297 , analogy between perfect gas and, 301 Radiation constant, Planck's, 304 — , Stefan's, 302 — , Wien's. 304 Radiation correction, 29 Radiation, diffuse, 301 — from the stars, 314 Radiation laws, Planck's law, 304 — , Stefan-Boltzmanft's law, ii Radiation, measurement of, 297 et seq. Radiation, passage through matter of. Radiation, pressure of, 300 — , properties and nature of, 279 — , temperature, 293 Radiation pyrometers, 307 et seq, liative equilibrium, 317 Radiometer, Crookcs', 298 i meters, 297 ei seq, Radiomicrometcr, 300 220 ', 221 i of the sped .■rq. —, adiabatic expansion method (Cle- ment and Desormes) , 53 — , experiments of Partington, SS — , method of Ruchardt, 56 — , table of results, 62 — , velocity of sound method, 57 Reaction turbines, 233 Rectilinear diameters, law of, 100 Rectilinear flow of heat, 182 Reflecting power, 288 128 — , characteristics of, 129 Refrigerating machine, absorption, 131 — s air compression, 133 — Bell-Coleman, 133 — , Carnol, 218 , efficiency of, 235 — , vapour compression, 127 ——j efficiency of, 235 Refrigeration, Chap. VI — , cascade process of, 135 — due to Pettier effect, 133 — , principles used in, 125 by adding a sal , 125 b i • der n duced i VH |i|i:i||, - by Joule Thomson expo Refrigci mi Refrigerating machine i : u i . i '.••: OOOltng, 141 . thermal, 179 :e thermometry, 13, 23, ■'<'•'■' , platinum, 13 engines, 218 — process, 216 cket, 234 Room 154 i mean square velocity, 74 Rowland's experiments, 66 n I icperiments, 56 ..Ivc, 206 Satellites artificial, 2 Saturated vapour, 111 — , density of, 123 — , specific heat of, 273 — , vapour pressure of, 112-117 Searle's apparatus for conductivity, ISO Secondary thermometers, 10, 27 . 228 Single-acting engine, 209 Slide valve, 209 Solar constant, definition of, 311 — , determination of, by absolute pyr- imeter, 312 Solidification of helium, 147 Solids, conductivity of, see Conducti- vity of solids Sound, velocity of, 57 et seq. Specific beat, definition of, — , difference between the two, 46, 245, — , l-:!n ■ '' - ~, methods of measurement. Chap, II — , negative, 274 Specific heats of gases, 45 et seq. Specific: heat of gases, determ'nati n (experimental) at constant pressure, 49 et seq. — , at constant volume, 51 et seq. ■ ! -cory Specifn heat of liquids, 31, 37-40 d, 37 — , method of mixtures, 29 — , stead j- -flow electric calorimeter, 38 heat of vapour, saturated, 273 — unsaturated, 61 Specific heat of solids, 30, 41 et seq. — , electrical method, 41-43 — v method of mixtures, 29 — i variation with temperature. Specific heat of superheated vapour, 61 • , 40 Spectra of stars, 292 . V — -, therm idynamic treatment of, 262, 272 States of .104 Steam calorimeter (Joly), 34 et seq. , 205 ei sr ; — , < 208 — i, modern, 211 — , New men's atnn spl &rj arts of, 208 ei . — , Savi py's, 206 — , single-acting, 209 Steam, enti — , expansive use of, 209 — total beat of ii jet, theory of, 230 , 229 mp ,23; uf Curtis, 232 — of Dc Laval, 231 — of Parsons, 233 — , reaction, 2 Stefati-iBoltzmaiui'a lav,-, 302 — , experimental verification of, 302 Stefan's constant, : : Stral here, 317 — , pressure did : n in, 318 Stuffing box, 209 Sublimation, 105 Sulphur boiling apparatus, 12 Sun, radiation from, 311 et seq. — , temperature of i iling, 111 h a ting, 111 , 61 259 360 SUJJJIiCT INDEX Temperature, absolute, 10, 249 — baths, 11 -, critical, 90, 95, 100 — — , table of, 136 Temperature, critical, value_ of, from ve.jj tlur Waals' equation, 95 — , effective, 153 — , definition of, 1 — , distribution of, in the atmosphere, 316 _ gradient, 319 — , high, measurement of, 306 — , low, measurement of, 22 „ — t production of, Chap. VI — of inversion, 145, 268 — of stars, 314 — of sun, 314 — radiation, 293 — scale, international, 24 — . standard, 11 — , underground, 188 — wave, wavelength of, 188 Theorem, Carnot's, 218 Theory, atomic, 70 — , kinetic, of matter, Chap. Ill — , molecular, 70 — of exchanges (Frevost's), 286 Thermal definition of, 23 Xhet wt Condu.- (the i"' . , — , ratio of electrical conductivity to., 190 Thermal exp slon banco, 179 Thermal state of a bod: i rmo-enuples, 18 et se<"!. _ Thermodynamic relationships (3 well), 260 ei — , first relation, 2 , application to change of freez- ing' point by pressure, 262 } application to liquid film, 264 _ second relation, 264 — , third and fourth relations, 266 Thermodynamic scale of temperature, 10, 249 Thermodynamica! variable, 237 Thermodynamics, application ot, to change of state, 262, 272 Thermodynamics, application of, to radiation, see Thermodynamics of radiation Thermodynamics, examples on, 271^ Thermodynamics, first Jaw of, 243, 259 — of evaporation, 263 — of fusion, 262 Thermodynamics of radiation, 293 et seq. Thermodynamics of refrigeration, 235 Thermodynamics of the atmosphere, 316 Thermodynamics, scope of, 237 — , second law of, 247 Thermodynamics, second law of, — ? Clausius J s enunciation of, 248 , Kelvin's enunciation of, 248 — , preliminary statement of, 243 — , scope of, 247 — statement of, in terms of entropy, 255 Thermo-electric thermometry, 18 et seq. Thermometer, alcohol, 2, 23 — , Beckmatm, 3 — , Callendar's compensated air, 6 — , chemical, 3 — , clinical, 3 ^ — , constant volume hydrogen, 7 — , gas, 4 et seq. — , liquid, 1-3 — , maximum and minimum, 3 — , mercury', 1 —j petroleum ether, 23 — , platinum, 13 et seq, —, secondary, 10, 27 — , standard gas, 7 — , standardisation of, 10 — , thermo-electric, 18 et seq. — , vapour pressure, 22, 23 _, weight, 167 Thertnometric conductivity, 182 Thcrrnometr] , Char*. I — , high temperature, 306 el seq. — , low temperature, 22 — , resistance, 13 et seq. — , thermo-electric, 18 *-'-' seq. Thermopile, 279 Thcrmo-regulator, 174 Thermostat, 174 Throttle valve, 210 Toluene thermostat, 174 Total heat of steam, 221 Total radiation pyrometers, 308 Transformation, adiabatic, 4% Transport phenomena, 82 Triple point, 275 — for water, 275 Tropopausc, 317 — height of, 317 Troposphere, 317 — , pressure distribution in, 318 — ■ temperature distribution in, 319 t'urliin U i :gy, 256 i: round temperature, 188 tjaii of heat, linn, production of high, 149 • lr:r Walls' equation of state, 91 . critical constants from, 95 — , deduction of, 91 -— , defects in, 96 — , discussion of, 9S — , Joule-Thomson effect from, 140, 268 — , methods of finding a and b, 92 Vaporisation, 104, 111 et seq, — , latent heat of. 117 see also Latent heat of vaporisation Vapour compression machine, 127 — , efficiency of, 235 Vapour density, 122 et seq. — of saturated vapour, 123 — , Victor Meyer's method, 123 Vapour pressure curve for water, 102, 321 — , discussion of results, 116 — , laws of, for mixture of liquids, 116 Vapour pressure measurements, dyna- mic or boiling point method, 114 — , static method, 113 Vapour pressure of water, 112 — over carved surfaces, 117 Vapour pressure thermometers, 22, 24 Velocity, average, 78 — , law of distribution of, 76 t probable square, 74 ity of sound, 57 et seq. Vibrational motion of diatomic mole- cules, 80 Viscosity, 83 — , discussion of the result, 84 W Water, boiling point of, 11 — equivalent, 28 — , expansion of, 171 — , freezing point of, on absolute scale, 10 — , specific heat of, 40 — vapour in the atmosphere, 319 , method uf condensing, 324 — , vapour pressure of, 320 Watt's double-acting engine, 208 — experimental condenser, 208 — governor, 210 — inventions, 207 et seq. Weight thermometer, 167 Wiedemann and Franz's law, 190 W ion's constant, 304 Wien r s displacement law, 304 — , experimental verification of, 306 Work, graphical representation of, 213 — obtained in isothermal and adiabatic expansion, 245 Working substance, 214 X-rays, 285 Zero, absolute, 10, 250